1.5 Exponents and introduction to logarithms questions for IB Maths AI

Find the value of $a$ and the value of $k$.

Hence write down the annual percentage increase in the number of trout.

Find the number of years after the trout are introduced until the model predicts there will be 5000 trout in the lake.

Find the value of $k$.

Show that $r=0.0796$, correct to three significant figures.

Find the temperature of the coffee 12 minutes after it is made.

Chloé likes to drink her coffee when its temperature is between 50 °C and 60 °C. Find the length of time for which the coffee is in this temperature range.

Explain why, according to the model, the coffee never cools to 21 °C.

Chloé makes a second cup of coffee at the same initial temperature, but puts it outside where the temperature is 8 °C. Assuming the value of $r$ is unchanged, write down the new model for the temperature of this cup of coffee.

Show that $k=\frac{\ln 2}{6}$.

The patient is given a dose of 200 mg every 8 hours.

Show that the fraction of the antibiotic that remains in the body 8 hours after a dose is $0.397$, correct to three significant figures.

Let $r=0.396850\dots$ and let $S_n$ mg be the amount of antibiotic in the body immediately after the $n$th dose.

(i) Find $S_2$.

(ii) Show that $S_n=\frac{200(1-r^n)}{1-r}$.

(i) Find the amount of antibiotic in the body immediately after the 6th dose.

(ii) Find the amount of antibiotic in the body immediately before the 7th dose.

In the long term, the amount immediately after a dose approaches a limiting value, $S$ mg.

Find $S$, and the long-term amount immediately before each dose.

The antibiotic is effective only when there are at least 120 mg in the body.

Find the number of the first dose after which the amount in the body never falls below 120 mg.

To make the antibiotic effective sooner, the doctor gives a larger first dose, $L$ mg, called a loading dose, followed by doses of 200 mg every 8 hours as before.

(i) Write down an expression, in terms of $L$ and $r$, for the amount in the body immediately after the second dose.

(ii) Find the value of $L$ for which this amount is equal to the value of $S$ found in part (e).

Amounts above 400 mg are harmful. The doctor considers changing the schedule to a dose of $D$ mg every 12 hours.

(i) Find the largest value of $D$ for which the long-term amount immediately after a dose does not exceed 400 mg.

(ii) Explain why this schedule would not be suitable.

Amira returns to doses of 200 mg, now given every $\tau$ hours.

Find the largest value of $\tau$ for which the long-term amount immediately before each dose is at least 120 mg.

State one limitation of Amira's model.

Simplify ${\displaystyle \frac{(3x^{2}y{)}^3}{9x^{-1}{y}^2}}$, giving your answer in the form $a{x}^p{y}^q$, where $a,p,q\in \mathbb{Z}$.

Write ${\displaystyle \frac{5}{x^4}}$ in the form $a{x}^n$, where $a,n\in \mathbb{Z}$.

Solve $3^x=81$.

Solve $4^{x+1}=64$.

Calculate the pH of the water.

Mia's fish need water with a pH from 6.5 to 7.5.

Find the corresponding range of values of $\left[{\text{H}}^{+}\right]$. Give the end points of the range in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

State whether the water in the aquarium is suitable for Mia's fish. Justify your answer.

Mia adds a treatment to the water that raises the pH by exactly 0.6.

(i) Find the number by which the hydrogen ion concentration is divided.

(ii) Find the new hydrogen ion concentration. Give your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

Mia also tests the treated water with a paper strip, which gives a pH of 7.2.

Taking the pH from part (d) as the exact value, find the percentage error in the reading from the paper strip.

Write the equation ${10}^4=10000$ in logarithmic form.

Given that $\ln x=2.5$, find the value of $x$.

Solve ${10}^x=450$.

Write down the value of ${\log }_{10}0.0005$.

Explain why ${\log }_{10}(-5)$ has no real value.