1.5 Exponents and introduction to logarithms

Learning objectives

• Use the laws of exponents with integer exponents to simplify numerical and algebraic expressions.

• Evaluate expressions with a zero exponent or a negative exponent, and write reciprocals as negative powers.

• Solve simple exponential equations by writing both sides as powers of the same base.

• State the meaning of a logarithm and convert between the forms $a^x = b$ and $x = \log_a b$.

• Explain why the argument of a logarithm must be positive.

• Evaluate common logarithms (base 10) and natural logarithms (base e) using technology.

• Solve equations of the form $a^x = b$ using technology, writing down the equation that was solved.

• Calculate and interpret values on logarithmic scales such as pH, decibels and the Richter scale.

Introduction

Repeated multiplication appears wherever a quantity grows or shrinks by the same factor again and again: money earning compound interest, a population growing by a fixed percentage, a radioactive sample losing half its atoms in each half-life. Exponents are the short way of writing repeated multiplication, and the laws of exponents are the rules for calculating with them. Logarithms work in reverse: from the result of a repeated multiplication, a logarithm gives the exponent, the number of times the base was used. This makes an unknown exponent something that can be calculated.

Logarithms also make very wide ranges of values manageable. The hydrogen-ion concentration of liquids, the intensity of sounds and the amplitude of earthquake waves each vary by factors of millions or more, so scientists report them on logarithmic scales (pH, decibels and the Richter scale) where each step of 1 stands for multiplication by 10.

Exponents

Repeated multiplication

The product $3 \times 3 \times 3 \times 3$ is written $3^4$ and read "3 to the power 4". The 3 is the number being multiplied, and the small raised 4 counts how many times it appears in the product. The value is $3^4 = 81$.

The raised number has several names: "exponent", "power" and "index" (plural "indices") all mean the same thing, and all three are in common use.

For a positive integer exponent the definition is a counting statement, and every law of exponents follows from counting factors. The definition makes no sense for an exponent of 0 or $-3$ (a product cannot have zero or minus three factors), so those exponents are given meanings later, chosen so that the same rules keep working.

Brackets and negative bases

An exponent applies only to the symbol or bracket written directly before it. This matters most with negative numbers and with coefficients in algebra.

A negative base raised to an even power gives a positive result, and raised to an odd power gives a negative result, because the minus signs cancel in pairs. In $-3^2$ the power is worked out first and the minus sign afterwards, following the usual order of operations. A GDC follows the same rule, so typing -3^2 returns $-9$; to square $-3$ the brackets must be typed.

The laws of exponents

Multiplying powers with the same base

Multiplying $2^3$ by $2^4$ means writing three factors of 2 followed by four more.

There are $3 + 4 = 7$ factors in total. The same counting works for any base and any positive integer exponents, which gives the first law: to multiply powers of the same base, add the exponents.

Dividing powers with the same base

Dividing $5^6$ by $5^2$ cancels two of the six factors of 5 in the numerator against the two in the denominator, leaving four.

The number of factors left is $6 - 2$, so to divide powers of the same base, subtract the exponents: $a^m \div a^n = a^{m-n}$ for $a \ne 0$. The base must not be 0, because division by zero is undefined.

A power of a power and a power of a product

Raising $4^3$ to the power 2 multiplies two copies of $4^3$, which is $4^3 \times 4^3 = 4^6$. In general $m$ factors of $a$, repeated $n$ times, gives $mn$ factors.

A power of a product spreads over every factor in the product: $(ab)^3 = ab \times ab \times ab = a^3b^3$, because multiplication can be done in any order. The same is true for a quotient. These rules are collected in one place because they are always used together.

The first three laws only apply when the bases are the same. The product $2^3 \times 5^2$ cannot be written as one power, and has to be evaluated as $8 \times 25 = 200$. Powers with different bases can sometimes be rewritten with a common base first: $4^3 = (2^2)^3 = 2^6$, so $4^3 \times 2^5 = 2^6 \times 2^5 = 2^{11}$.

YouTube

The zero exponent

The quotient law gives a value for an exponent of 0. Dividing any non-zero power by itself gives 1, and subtracting the exponents gives an exponent of 0.

The only way for both statements to be true is $a^0 = 1$. The pattern in a table of powers of 2 leads to the same value: each step down the table halves the result, and halving $2^1 = 2$ gives 1.

The quotient argument cannot give a value for $0^0$, because it would need division by 0, so $0^0$ is left undefined.

Negative exponents

Continuing the table past $2^0$ halves the value at each step, which gives $2^{-1} = \frac{1}{2}$, $2^{-2} = \frac{1}{4}$ and $2^{-3} = \frac{1}{8}$. The quotient law gives the same result: $\frac{2^2}{2^5} = 2^{2-5} = 2^{-3}$, while cancelling factors gives $\frac{2 \times 2}{2 \times 2 \times 2 \times 2 \times 2} = \frac{1}{2^3}$. A negative exponent therefore means "one divided by the positive power".

With this word, a negative exponent has a short description: $a^{-n}$ is the reciprocal of $a^n$.

The second form follows from the first: $\left(\frac{a}{b}\right)^{-n} = 1 \div \frac{a^n}{b^n} = \frac{b^n}{a^n}$. To raise a fraction to a negative power, turn the fraction upside down and use the positive power, so $\left(\frac{2}{5}\right)^{-2} = \left(\frac{5}{2}\right)^2 = \frac{25}{4}$. A negative exponent never makes a number negative: $2^{-3}$ is the small positive number $0.125$.

Negative exponents are already familiar from standard form in subtopic 1.1, Scientific notation: $4.7 \times 10^{-6}$ means $4.7 \times \frac{1}{1\,000\,000} = 0.000\,0047$. Each step down in the power of 10 divides by 10.

Using the laws with algebra

The laws work for letters in the same way as for numbers. With several letters and coefficients, deal with the numbers first, then each letter in turn, using one law at a time. Writing a power of a product as a product of powers before simplifying keeps the steps clear.

Writing expressions such as $\frac{5}{x^2}$ in the form $5x^{-2}$ prepares them for differentiation and integration in topic 5, where every term has to be written as a multiple of a power of $x$.

Substituting small numbers, as in that check, is a quick way to catch a sign slip with negative exponents, a frequent error in this kind of simplification.

Exponential equations

Equations with the unknown in the exponent

An equation such as $2^x = 32$ asks for an exponent rather than a base. The unknown sits in the exponent, so the methods for linear or quadratic equations do not apply directly.

When both sides can be written as powers of the same base, the exponents must be equal. This works because, for a positive base other than 1, different exponents always give different values: the powers of 2 are 1, 2, 4, 8, 16, 32, …, and only one exponent gives 32. So $2^x = 32 = 2^5$ gives $x = 5$.

This matching method works only when the numbers happen to be powers of a common base. Most equations from real situations are not like that. A sum of money invested at 5% compound interest a year doubles when $1.05^x = 2$, and 2 is not a whole-number power of 1.05: $1.05^{14} = 1.98\ldots$ and $1.05^{15} = 2.07\ldots$, so the answer lies between 14 and 15. Finding it exactly needs either technology or a new kind of number, the logarithm.

Solving exponential equations with the GDC

The GDC solves $a^x = b$ in two standard ways. The menu names for two common models are given as examples; other models have equivalent tools.

1. Graph and intersect. Enter the left side as one function and the right side as another ($Y_1 = 1.05^x$ and $Y_2 = 2$), choose a window that shows the crossing, and use the intersection tool (TI-84 Plus CE: 2nd TRACE → intersect; Casio fx-CG50: SHIFT F5 (G-SOLVE) → INTSECT). The $x$-coordinate of the point of intersection is the solution.

2. Equation solver. Enter $1.05^x = 2$ (or $1.05^x - 2 = 0$) in the numerical solver (TI-84 Plus CE: MATH → Numeric Solver; Casio fx-CG50: MENU → Equation → SOLVER) with a sensible starting value, and solve for $x$.

The graph of $y = 1.05^x$ rises steadily, so it crosses the line $y = 2$ exactly once, at $x \approx 14.2$.

The solution of $1.05^x = 2$ is the doubling time for 5% growth, about 14.2 years, a quantity met in subtopic 1.3, Geometric sequences and series, and used with compound interest in subtopic 1.4.

In the final step of that example the GDC returns 8.91, while the question asks for a whole number of years. Rounding to the nearest integer would give the wrong answer for a value such as 8.3, because the population would still pass 25,000 only during the ninth year. Whether to round up or down depends on what the question asks.

Logarithms

The definition

The number of times a base must be multiplied to give a result is called a logarithm. Since $10^3 = 1000$, the logarithm of 1000 in base 10 is 3; since $2^5 = 32$, the logarithm of 32 in base 2 is 5. The logarithm is the exponent, written as the answer rather than as a raised number.

The two forms are two ways of writing one fact, an example of equivalence. The exponential form $a^x = b$ is the natural way to calculate $b$ from $x$; the logarithmic form $x = \log_a b$ is the natural way to calculate $x$ from $b$. A useful way to read $\log_a b$ aloud is "the power of $a$ that gives $b$".

Why the base and the argument have conditions

The base $a$ is restricted for practical reasons. A base of 1 is useless, because every power of 1 equals 1, so $\log_1 5$ would ask for a power of 1 that equals 5, and there is none. A negative base gives powers that jump between positive and negative values and are undefined for many exponents, so negative bases are excluded too.

The number $b$ whose logarithm is taken must be positive. For a positive base, every power $a^x$ is positive, whatever the value of $x$: $10^3 = 1000$, $10^0 = 1$ and $10^{-3} = 0.001$ are all greater than zero, and very negative exponents only bring the value closer to zero without reaching it. There is therefore no exponent that turns 10 into 0 or into $-5$, and $\log_{10} 0$ and $\log_{10}(-5)$ do not exist. A GDC shows an error (or a non-real result) for them.

Logarithms and exponentials undo each other

Because a logarithm finds an exponent, taking the base-$a$ logarithm of $a^x$ simply returns $x$, and raising $a$ to the power $\log_a x$ returns $x$. For base 10:

The functions $x \mapsto 10^x$ and $x \mapsto \log_{10} x$ are inverse functions: each one reverses the other, in the same way that squaring and taking the positive square root reverse each other for positive numbers. The graph of $y = \log_{10} x$ is the reflection of the graph of $y = 10^x$ in the line $y = x$.

The reflection explains three features of the logarithm graph. It passes through $(1, 0)$ because $10^0 = 1$. It exists only for $x > 0$, because $10^x$ only produces positive values. It rises without limit but very slowly: $\log_{10} x$ reaches 1 at $x = 10$, 2 at $x = 100$ and only 6 at $x = 1\,000\,000$. That slow growth is what makes logarithms useful for describing quantities that range over many orders of magnitude.

Common logarithms

Base 10

Logarithms in base 10 fit the decimal number system, because the logarithm of a power of 10 is just the exponent. Printed tables of base-10 logarithms were the standard aid to calculation for more than three centuries, and they are the logarithms used by the pH, decibel and Richter scales.

When no base is written, $\log x$ means $\log_{10} x$, which is also what the GDC's LOG key calculates.

Estimating a common logarithm

For numbers that are not exact powers of 10, the common logarithm lies between two integers. Since $100 < 500 < 1000$, $\log 500$ lies between $\log 100 = 2$ and $\log 1000 = 3$. The GDC gives $\log 500 = 2.698\ldots$. The integer part, 2, comes from the order of magnitude of 500, and the decimal part comes from the 5.

This links logarithms to standard form. A number written as $a \times 10^k$ with $1 \le a < 10$ has a common logarithm between $k$ and $k + 1$, because $10^k \le a \times 10^k < 10^{k+1}$. For example, $7.2 \times 10^6$ has $\log$ between 6 and 7 (the GDC gives 6.857), and $4 \times 10^{-3}$ has $\log$ between $-3$ and $-2$ (the GDC gives $-2.398$). The integer below the logarithm is the order of magnitude of the number.

Euler's number and the natural logarithm

The number e

One number other than 10 is used as a base so often that it has its own letter. It appears when interest is added more and more frequently. An investment of 1 at an annual rate of 100%, compounded $n$ times a year, grows in one year to $\left(1 + \frac{1}{n}\right)^n$. Compounding more often gives more, but the extra gained shrinks each time.

The values settle towards 2.718 281 828…, a number that cannot be written as a fraction or as a terminating or repeating decimal. It is called e. Interest added at every instant in this way is continuous compounding, met in subtopic 1.4.

Beyond interest rates, e is the base for which growth is simplest to describe with calculus, so it appears in models of population growth, radioactive decay, cooling and the spread of disease. Models of the form $y = k\mathrm{e}^{rt}$ are met in subtopic 2.5 and, at HL, in subtopic 2.9.

The natural logarithm

A logarithm with base e is called a natural logarithm. It has its own notation and its own key.

Natural logarithms behave in exactly the same way as common logarithms; only the base differs. Since e is smaller than 10, it takes a larger power of e than of 10 to reach a given number, so $\ln x$ is larger than $\log x$ for $x > 1$: $\ln 20 = 2.996$ while $\log 20 = 1.301$.

Evaluating logarithms with technology

The LOG and LN keys

Logarithms of most numbers cannot be found by hand, so they are evaluated with the GDC. Every GDC has a LOG key for base 10 and an LN key for base e, with $10^x$ and $\mathrm{e}^x$ as their second functions, because each pair are inverses. Most models open a bracket after the function name, and the whole argument must go inside it before the bracket is closed.

• $\log(3.5 \times 10^4)$: type log(3.5×10^4) or log(3.5E4) and close the bracket: 4.544.

• $\ln\left(\frac{8}{3}\right)$: type ln(8/3): 0.981. Without the bracket around $\frac{8}{3}$, some models calculate $\frac{\ln 8}{3} = 0.693$ instead.

• $-\log(2.4 \times 10^{-9})$: type the minus sign first, then log(2.4E-9): 8.620.

Logarithms in other bases

The definition holds for any base $a > 0$, $a \ne 1$, although base 10 and base e are the ones in everyday use. Many GDCs have a logarithm template or a logBASE function that accepts any base directly, which gives $\log_{1.05} 2 = 14.2067\ldots$, the doubling time found earlier. On a calculator without one, $\frac{\ln b}{\ln a}$ gives the same value as $\log_a b$ (or $\frac{\log b}{\log a}$); the reason for this comes from the laws of logarithms (HL subtopic 1.9). Solving $a^x = b$ by a graph or the solver avoids the issue altogether.

Solving $a^x = b$ with a logarithm

The definition turns any equation $a^x = b$ into a calculation: $x = \log_a b$. For base 10 and base e the GDC evaluates this directly, so $10^x = 750$ gives $x = \log 750 = 2.875$, and $\mathrm{e}^x = 0.2$ gives $x = \ln 0.2 = -1.609$. For other bases, a graph, the solver or a log template is used, and the equation solved is written down.

The coefficient in front of the power must be removed before the definition is used. In (b), writing $x = \ln 30 \div 4$ or $x = \ln(30) - 4$ are both wrong, because $4\mathrm{e}^x$ is not a power of e until the 4 is divided out.

YouTube

Logarithmic scales

Why logarithmic scales are used

Some quantities vary over such a wide range that an ordinary number line cannot show them usefully. On a linear scale from 0 to 1,000,000, the values 1, 10, 100 and 1000 are squeezed into the first tiny fraction of the line. On a logarithmic scale, each step of the same length multiplies the value by the same factor, usually 10, so 1, 10, 100, 1000, … are equally spaced.

The comparison of the two scales shows that the position on a logarithmic scale is the logarithm of the value: 1000 sits at position 3 because $\log 1000 = 3$. Equal distances therefore stand for equal ratios rather than equal differences.

Logarithmic scales are a choice of representation. They let quantities from the very small to the very large be compared on one scale, and they roughly match the way human hearing and sight respond to ratios of intensity rather than differences. Their cost is that a small-looking change in the scale value can be a large change in the quantity, which makes reports of pH, sound level and earthquakes easy to misread.

The pH scale

Chemists measure the acidity of a solution by the concentration of hydrogen ions in it, written $[\mathrm{H}^+]$ and measured in moles per litre (mol/L). Concentrations in everyday liquids range from about 1 mol/L in strong acids to about $10^{-14}$ mol/L in strong alkalis, a factor of 100 trillion. The pH scale replaces these concentrations by small numbers.

The minus sign makes the pH positive, because the concentrations are less than 1 and so have negative logarithms. It also reverses the direction of the scale: a higher concentration of hydrogen ions (a more acidic solution) gives a lower pH. The second form comes straight from the definition of a logarithm and is used to find a concentration from a pH.

A coffee with hydrogen-ion concentration $1.3 \times 10^{-5}$ mol/L has pH $-\log_{10}(1.3 \times 10^{-5}) = 4.886\ldots \approx 4.89$. A liquid with ten times that concentration, $1.3 \times 10^{-4}$ mol/L, has pH $-\log_{10}(1.3 \times 10^{-4}) = 3.886\ldots \approx 3.89$. Ten times the concentration lowers the pH by exactly 1, and evaluating other pairs gives the same difference every time; the reason is a law of logarithms (HL subtopic 1.9).

Case study: ocean acidification

The surface of the ocean absorbs part of the carbon dioxide released by burning fossil fuels. Dissolved carbon dioxide forms carbonic acid, which raises the hydrogen-ion concentration of seawater and lowers its pH. Seawater is slightly alkaline, with a pH a little above 8, and since the Industrial Revolution the global average pH of the surface ocean has fallen by about 0.11.

A fall of 0.11 sounds negligible, but pH is a logarithmic scale, so the change has to be translated back into concentrations. Taking a starting pH of 8.20 for illustration, the pH after the fall is 8.09.

Source: change of 0.11 in global average surface pH since the Industrial Revolution, NOAA Pacific Marine Environmental Laboratory, "A primer on pH", accessed 2026. The starting value 8.20 is illustrative.

The ratio of the concentrations is $\frac{8.128 \times 10^{-9}}{6.310 \times 10^{-9}} = 1.288$, so the hydrogen-ion concentration has risen by about 28.8%. Repeating the calculation from a different starting pH, such as 7.71 falling to 7.60, gives the same ratio 1.288, because the ratio depends only on the size of the fall: it equals $10^{0.11}$. NOAA rounds this to "approximately a 30% increase in acidity".

The case shows why reports on logarithmic scales need care. A fall of 0.11 pH units is an increase of almost 30% in hydrogen-ion concentration; a fall of a full unit would be a tenfold increase. The model also has limits. pH varies from place to place and from season to season by more than 0.11, so a single global average hides local conditions, and the definition with concentration is itself an approximation that chemists refine for seawater.

The Richter scale

Earthquakes release energy across an enormous range, and the seismograph traces they produce vary in amplitude (the size of the largest swing) by many factors of 10. In 1935 Charles Richter, working with Beno Gutenberg, defined a magnitude scale for earthquakes in southern California based on the logarithm of this amplitude.

Rewriting the definition in exponential form gives the amplitude from the magnitude: $\frac{A}{A_0} = 10^M$, so $A = A_0 \times 10^M$. Two earthquakes of magnitudes $M_1$ and $M_2$ therefore have amplitudes in the ratio $\frac{A_0 \times 10^{M_1}}{A_0 \times 10^{M_2}} = 10^{M_1 - M_2}$, by the quotient law of exponents. A difference of 2 in magnitude means 100 times the amplitude; a difference of 3 means 1000 times.

Seismologists now use the moment magnitude scale for large earthquakes, because Richter's original scale saturates: above about magnitude 7 its values stop rising in step with the size of the earthquake, but the values are close for moderate earthquakes and news reports often still say "Richter scale". The energy released also grows much faster than the amplitude, a relationship studied with the laws of logarithms in HL subtopic 1.9.

The decibel scale

Sound intensity $I$, the power carried by a sound wave through each square metre, ranges from about $10^{-12}$ W/m² for the faintest sound a typical person can hear to more than 1 W/m² for sounds that cause pain. The sound intensity level compresses this range of a trillion or more into numbers from about 0 to about 120.

The factor 10 in front of the logarithm is there because the original unit, the bel, was felt to be too large; a decibel is one tenth of a bel. Since $\log_{10}$ increases by 1 for each factor of 10, multiplying the intensity by 10 adds $10 \times 1 = 10$ dB, and multiplying by 100 adds 20 dB.

Exam focus

Exponents and logarithms are examined mainly on Paper 1 in short questions of 2–6 marks, and appear as single parts of longer Paper 2 questions on financial mathematics or exponential models. Three question shapes recur.

• Simplifying with the laws of exponents, usually 2–3 marks with simplify, write or express ("express in the form $ax^n$"). Every step should be shown, one law at a time.

• Evaluating a logarithmic formula in context: pH, sound level, earthquake magnitude, or a formula defined in the question. Calculate earns (M1) for the correct substitution, written with the whole argument inside the logarithm, and A1 for the answer to 3 significant figures.

• Rearranging to the exponential form or solving $a^x = b$: "find the concentration", "find the time when …". Markschemes award a method mark for an exponential equation with the correct base ($C = 10^{-\text{pH}}$), or for the equation entered in the GDC.

Justify and determine whether questions ask for a conclusion supported by calculation. Comparing two inputs (two concentrations) does not answer a question about the outputs (two pH values); the outputs must be calculated or a reason given that links one to the other.

Part (d) is answered by evaluating both levels and subtracting. HL students prove that doubling any intensity adds $10\log_{10} 2 \approx 3.01$ dB with a law of logarithms (subtopic 1.9).

When an answer comes from a power of 10, give it as a decimal or in standard form unless an exact form is requested. Leaving $I = 10^{-1.5}$ when the question asks for a value can lose the final mark; markschemes give the decimal, 0.0316.

Practice questions

1. Simplify $\dfrac{6^5 \times 6^{-2}}{6^4}$, giving your answer as a power of 6. [2]

2. Write $\dfrac{7}{2x^3}$ in the form $ax^n$. [2]

3. Simplify $\dfrac{(3p^2q^{-1})^3}{9p^4q^2}$, giving your answer without negative exponents. [3]

4. Solve $9^{x} = 3^{x+4}$. [3]

5. (a) Write $\log_{10} 0.0001 = -4$ in exponential form. [1] (b) Find the exact value of $x$ if $\ln x = 3$, and give its value to 3 significant figures. [2]

6. Use technology to evaluate $\log(4.6 \times 10^{-7})$ and $\ln 250$, correct to 3 significant figures. [2]

7. A car worth $28,000 loses 15% of its value each year, so its value after $t$ years is $V = 28\,000 \times 0.85^t$. Find the number of complete years after which the car is first worth less than $10,000. [3]

8. Rainwater in a city has pH 4.6, and rainwater in a mountain village has pH 5.6. Use $\text{pH} = -\log_{10} C$. (a) Find the hydrogen-ion concentration $C$ of the city's rainwater. [2] (b) Find how many times greater the city's concentration is than the village's. [2]

9. Earthquake A has magnitude 7.3 on the Richter scale. Earthquake B produces seismograph waves with one-fiftieth of the amplitude of A's. Find the magnitude of B. [3]

10. A sound has level 72 dB, where $L = 10\log_{10}\left(\dfrac{I}{10^{-12}}\right)$. (a) Find its intensity $I$. [2] (b) A second sound is 1000 times as intense. Write down its level. [1]

Summary

• $a^n$ means $n$ factors of the base $a$; an exponent applies only to the symbol or bracket directly before it, so $(2x)^4 = 16x^4$, while in $2x^4$ only $x$ is raised to the power 4.

• Laws of exponents (same base): $a^m \times a^n = a^{m+n}$, $a^m \div a^n = a^{m-n}$, $(a^m)^n = a^{mn}$; also $(ab)^n = a^nb^n$.

• $a^0 = 1$ and $a^{-n} = \dfrac{1}{a^n}$ for $a \ne 0$; a negative exponent gives a reciprocal, never a negative number.

• An exponential equation can be solved by matching bases when both sides are powers of one number; otherwise use logarithms or the GDC (graph intersection or solver), writing down the equation solved.

• $a^x = b \Leftrightarrow x = \log_a b$ with $a > 0$, $a \ne 1$, $b > 0$: a logarithm is an exponent, and the logarithm of zero or a negative number does not exist.

• $\log x$ means $\log_{10} x$; $\ln x = \log_{\mathrm{e}} x$, where $\mathrm{e} = 2.718\ldots$ is Euler's number. Each is the inverse of $10^x$ or $\mathrm{e}^x$.

• Logarithms are evaluated with the LOG and LN keys, with the whole argument in brackets; estimate first from the power of 10.

• On a logarithmic scale each step of 1 stands for a factor of 10: pH $= -\log_{10}[\mathrm{H}^+]$, Richter $M = \log_{10}\frac{A}{A_0}$, decibels $L = 10\log_{10}\frac{I}{I_0}$ (10 dB per factor of 10).

Self-check

• To do:

  I can identify the base and the exponent in a power and use brackets correctly with negative numbers and coefficients.

• To do:

  I can justify each law of exponents by counting factors.

• To do:

  I can simplify numerical and algebraic expressions using the laws of exponents one step at a time.

• To do:

  I can explain why $a^0 = 1$ and evaluate negative powers, including powers of fractions.

• To do:

  I can write expressions such as $\frac{5}{x^2}$ in the form $ax^n$.

• To do:

  I can solve exponential equations by writing both sides with the same base.

• To do:

  I can solve $a^x = b$ with the GDC by graph intersection or the solver, and write down what I did.

• To do:

  I can convert between $a^x = b$ and $x = \log_a b$ in both directions.

• To do:

  I can explain why the logarithm of zero or a negative number is undefined.

• To do:

  I can evaluate $\log$ and $\ln$ with the GDC and check the answer with an estimate.

• To do:

  I can explain where the number e comes from and what $\ln x$ means.

• To do:

  I can calculate a pH, a Richter magnitude or a decibel level from a formula, and rearrange to find the underlying quantity.

• To do:

  I can compare two values on a logarithmic scale by converting the difference back into a factor.

• To do:

  I can give an answer that comes from a power of 10 as a decimal and round a time in context in the right direction.