Exponential and Logarithmic Functions: math practice, Grade 12 – download the PDF

1 Evaluating an exponential function ★★★

Let \( f(x)=3\cdot 2^{x} \). Compute \( f(0) \), \( f(4) \) and \( f(-1) \).

2 Growth or decay? ★★★

For each function, say whether it models growth or decay and give the percent change per step: (a) \( y=5(1.08)^{x} \) (b) \( y=200(0.9)^{x} \) (c) \( y=7\left(\tfrac32\right)^{x} \) (d) \( y=4\left(\tfrac13\right)^{x} \).

3 From exponential to logarithmic form ★★★

Rewrite each equation in logarithmic form: (a) \( 2^{7}=128 \) (b) \( 10^{-2}=0.01 \) (c) \( 9^{1/2}=3 \).

4 Logarithms without a calculator ★★★

Evaluate: (a) \( \log_3 81 \) (b) \( \log_5\dfrac1{25} \) (c) \( \log_4 2 \) (d) \( \ln e^{5} \) (e) \( \log 1000 \).

5 Condensing logarithms ★★★

Write each as a single number: (a) \( \log_6 4+\log_6 9 \) (b) \( \log_2 40-\log_2 5 \) (c) \( 2\log_3 6-\log_3 4 \).

6 Matching bases ★★★

Solve: (a) \( 2^{x}=64 \) (b) \( 5^{x+1}=125 \) (c) \( 10^{x}=0.001 \) (d) \( \left(\tfrac12\right)^{x}=8 \).

7 Town population ★★★

A town of 12,000 people grows 3% per year. (a) Write a formula for the population \( P(t) \) after \( t \) years. (b) Find the population after 2 years, rounded to a whole person.

8 Simple logarithmic equations ★★★

Solve: (a) \( \log_3 x=4 \) (b) \( \log_2(x-1)=3 \) (c) \( \ln x=0 \) (d) \( \log_x 49=2 \) with \( x\gt0 \).

9 Using the change of base formula ★★★

Use a calculator and the change of base formula to approximate to four decimal places: (a) \( \log_5 40 \) (b) \( \log_2 10 \).

10 An exponential equation with a calculator ★★★

Solve \( 7^{x}=30 \). Give the exact answer and a four-decimal approximation.

11 Equation with base e ★★★

Solve \( 4e^{2x}=36 \).

12 A logarithmic equation ★★★

Solve \( \log_4(3x+1)=2 \) and verify your answer.

13 Monthly compounding ★★★

You deposit $2,500 at 4.8% annual interest compounded monthly. How much will you have after 5 years?

14 When does the car lose enough value? ★★★

A car bought for $24,000 loses 15% of its value each year, so \( V(t)=24000(0.85)^{t} \). After how many whole years is it worth less than $10,000?

15 Medicine in the body ★★★

A patient takes 80 mg of a medicine whose half-life is 6 hours, so \( N(t)=80\left(\tfrac12\right)^{t/6} \) (see the graph).

(a) How many mg remain after 10 hours? (b) After how many hours are 10 mg left?

16 Expanding a logarithm ★★★

Expand \( \ln\dfrac{x^{3}\sqrt y}{e^{2}} \) using the laws of logarithms, for \( x,y\gt0 \).

17 Equation in quadratic form ★★★

Solve \( 9^{x}-4\cdot3^{x}-45=0 \).

18 Two logs, one extraneous root ★★★

Solve \( \log x+\log(x-21)=2 \).

19 Equating arguments ★★★

Solve \( \ln(x+2)+\ln(x-2)=\ln5 \).

20 Doubling and tripling time ★★★

An account earns interest compounded continuously. (a) At 4.5%, how long to double? (b) At 6%, how long to triple? Round to two decimals.

21 Chemistry: pH ★★★

The pH of a solution is \( \text{pH}=-\log[\text{H}^{+}] \), where \( [\text{H}^{+}] \) is the hydrogen ion concentration in moles per liter. (a) Find the pH when \( [\text{H}^{+}]=3.2\times10^{-5} \). (b) Blood has pH 7.4: find its concentration.

22 Fitting a model to two data points ★★★

A culture has 800 cells at \( t=0 \) hours and 5,000 cells at \( t=4 \) hours. Assume \( N(t)=800e^{kt} \). (a) Find \( k \). (b) Predict the number of cells at \( t=6 \).

23 True or false? ★★★

Decide whether each statement is true or false, and justify with a counterexample or a law. (a) \( \log(a+b)=\log a+\log b \) for all \( a,b\gt0 \). (b) \( \ln(x^{2})=2\ln x \) for every real \( x\neq0 \). (c) \( \log_b x\cdot\log_x b=1 \) for \( b,x\gt0 \), \( b,x\neq1 \).

24 Annual versus continuous ★★★

You invest $800 at 7%. (a) If interest is compounded annually, what is the first whole year in which the balance reaches $5,000? (b) How long does it take with continuous compounding?