Grade 12 · MHF4U · Advanced Functions

MHF4U: Exponential and Log Word Problems

The word problems in this unit come in two kinds. Growth and decay problems use an exponential model and usually ask how long something takes, which needs a logarithm. Log scale problems, such as pH and earthquake magnitude, use a logarithm to squeeze huge ranges into small numbers.

How It Works

Doubling and half-life. A quantity that doubles every $p$ units of time is $A = A_0 \cdot 2^{t/p}$ , and one with half-life $p$ is $A = A_0 \left(\frac{1}{2}\right)^{t/p}$ . The exponent $\frac{t}{p}$ counts how many doublings or halvings have happened, so after $3$ half-lives one eighth of the starting amount remains.

Percent growth and compound interest. Growth of $r$ per period is $A = A_0 (1 + r)^n$ , and decay is $A = A_0 (1 - r)^n$ . Interest at an annual rate $i$ compounded $m$ times a year is $A = P\left(1 + \frac{i}{m}\right)^{mt}$ , with the rate divided and the time multiplied by $m$ .

Solving for the time. Divide both sides by the starting amount first, so the power is alone. Then take the log of both sides and use the power law to bring the exponent down. Round only the final answer, and state it in the units the question asks for.

Log scales. $\text{pH} = -\log[\text{H}^+]$ , where $[\text{H}^+]$ is the hydrogen ion concentration. On the Richter scale, each step of $1$ is ten times the intensity, so two earthquakes whose magnitudes differ by $M$ compare as $10^M$ . The decibel scale works the same way with $10$ dB for each factor of ten. These questions are about ratios, so subtract the log values first, then raise $10$ to the difference.

5 Worked Examples

Easy first, test level last. Try each one on paper, then open the solution.

Easy
A solution has $[\text{H}^+] = 3.2 \times 10^{-4}$ mol/L. Find its pH.
Show solution
1. $\text{pH} = -\log(3.2 \times 10^{-4})$ .
2. $\log(3.2 \times 10^{-4}) = \log 3.2 - 4 \approx 0.505 - 4 = -3.495$ .
Answer: pH $\approx 3.49$
Easy
A culture of $500$ bacteria doubles every $20$ minutes. How many bacteria are there after $2$ hours?
Show solution
1. $2$ hours is $120$ minutes, which is $\frac{120}{20} = 6$ doublings.
2. $N = 500 \cdot 2^6 = 500 \cdot 64$ .
Answer: $32\,000$ bacteria
Medium
A patient is given $200$ mg of a drug with a half-life of $6$ hours. How long until $30$ mg is left?
Show solution
1. $200 \left(\frac{1}{2}\right)^{t/6} = 30$ , so $\left(\frac{1}{2}\right)^{t/6} = 0.15$ .
2. Take logs: $\frac{t}{6} \log 0.5 = \log 0.15$ .
3. $t = \frac{6 \log 0.15}{\log 0.5} \approx 6(2.737) \approx 16.4$ .
Answer: About $16.4$ hours
Medium
How many times as intense is an earthquake of magnitude $7.2$ as one of magnitude $5.4$ on the Richter scale?
Show solution
1. The magnitudes differ by $7.2 - 5.4 = 1.8$ .
2. Each step of $1$ is a factor of $10$ , so the ratio is $10^{1.8} \approx 63.1$ .
Answer: About $63$ times as intense
Test level
$5000$ dollars is invested at $4.5\%$ per year, compounded monthly. How long until it is worth $8000$ dollars?
Show solution
1. Monthly rate $\frac{0.045}{12} = 0.00375$ ; after $n$ months $A = 5000(1.00375)^n$ .
2. $5000(1.00375)^n = 8000$ , so $(1.00375)^n = 1.6$ .
3. $n = \frac{\log 1.6}{\log 1.00375} \approx 125.6$ , so the balance passes $8000$ dollars at the $126$ th month.
4. $126$ months is $10.5$ years. Check: $5000(1.00375)^{126} \approx 8013$ , and $5000(1.00375)^{125} \approx 7983$ .
Answer: $126$ months, or $10.5$ years

Learn It on Martian Lab

Martian Lab teaches this in the Applications of Logarithmic & Exponential Functions lesson of its MHF4U Exponential & Logarithmic Functions unit. Watch it, then take the Exponential & Logarithmic Functions practice tests to see if it stuck.

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Free Exponential & Logarithmic Functions practice test, with solutions

More Exponential & Logarithmic Functions topics: Laws of Logarithms · Solving Logarithmic Equations · Solving Exponential Equations · Graphing Logarithmic Functions

More MHF4U topics: Long and Synthetic Division · Vertical and Horizontal Asymptotes · Proving Trig Identities · Solving Trig Equations · End Behaviour and Degree · Finding the Equation of a Polynomial · Even and Odd Functions · Remainder and Factor Theorem · Solving Polynomial Equations · Polynomial Inequalities · Sum and Difference of Cubes · Holes in Rational Functions · Graphing Rational Functions · Rational Equations and Inequalities · Radian Measure · Unit Circle and Special Angles · Compound Angle Formulas · Double Angle Formulas · Graphing Sine and Cosine with Transformations · Graphs of Tan, Sec, Csc and Cot · Sinusoidal Word Problems · Sums, Products and Quotients of Functions · Composition of Functions · Average vs Instantaneous Rate of Change

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