Solve $5^x = 40$. Give the exact answer and a decimal to three places.

$x = \frac{\log 40}{\log 5} \approx 2.292$

Solve $3^{2x - 1} = 5^{x + 2}$ to three decimal places.

$x = \frac{\log 75}{\log 1.8} \approx 7.345$

Solve $4^x - 2^{x + 1} - 15 = 0$.

$x = \frac{\log 5}{\log 2} \approx 2.322$

Grade 12 · MHF4U · Advanced Functions

MHF4U: Solving Exponential Equations

An exponential equation has the unknown in the exponent, as in $2^{x + 3} = 32$ or $5^x = 40$ . Getting $x$ down from the exponent takes one of two moves: write both sides as powers of the same base, or take the logarithm of both sides.

How It Works

When both sides are powers of one base, make the bases match. $4^{x + 1} = 8^{x - 1}$ becomes $2^{2x + 2} = 2^{3x - 3}$ , and two equal powers of the same base have equal exponents: $2x + 2 = 3x - 3$ . This gives an exact answer with no calculator.

When they are not, take the log of both sides. The power law $\log(a^n) = n \log a$ brings the exponent down. From $5^x = 40$ you get $x \log 5 = \log 40$ , so $x = \frac{\log 40}{\log 5}$ . That fraction is the exact answer; round only at the end. Either $\log$ or $\ln$ works, as long as you use the same one on both sides.

When $x$ is on both sides with different bases, take logs, expand the brackets, and collect the $x$ terms on one side like any linear equation. The coefficients are logs, but they are just numbers.

When there are two terms with the unknown, look for a quadratic. $4^x$ is $(2^x)^2$ , so $4^x - 2^{x + 1} - 15 = 0$ is $u^2 - 2u - 15 = 0$ with $u = 2^x$ . Factor, then go back to $2^x$ . A power of a positive base is always positive, so a negative value of $u$ gives no solution and is rejected.

Check each answer in the original equation. It catches a rejected root you forgot to reject and a rounding slip.

5 Worked Examples

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

Easy
Solve $2^{x + 3} = 32$ .
Show solution
1. $32 = 2^5$ , so $2^{x + 3} = 2^5$ .
2. Equal bases, so equal exponents: $x + 3 = 5$ .
Answer: $x = 2$
Easy
Solve $5^x = 40$ . Give the exact answer and a decimal to three places.
Show solution
1. $40$ is not a power of $5$ , so take the log of both sides: $x \log 5 = \log 40$ .
2. $x = \frac{\log 40}{\log 5} \approx 2.292$ .
3. Check: $5^{2.292} \approx 40.0$ .
Answer: $x = \frac{\log 40}{\log 5} \approx 2.292$
Medium
Solve $4^{x + 1} = 8^{x - 1}$ .
Show solution
1. Write both as powers of $2$ : $(2^2)^{x + 1} = (2^3)^{x - 1}$ , so $2^{2x + 2} = 2^{3x - 3}$ .
2. $2x + 2 = 3x - 3$ , so $x = 5$ .
3. Check: $4^6 = 4096$ and $8^4 = 4096$ .
Answer: $x = 5$
Medium
Solve $3^{2x - 1} = 5^{x + 2}$ to three decimal places.
Show solution
1. Take logs: $(2x - 1) \log 3 = (x + 2) \log 5$ .
2. Expand: $2x \log 3 - \log 3 = x \log 5 + 2 \log 5$ .
3. Collect the $x$ terms: $x(2 \log 3 - \log 5) = 2 \log 5 + \log 3$ .
4. $x = \frac{2 \log 5 + \log 3}{2 \log 3 - \log 5} = \frac{\log 75}{\log 1.8} \approx 7.345$ .
Answer: $x = \frac{\log 75}{\log 1.8} \approx 7.345$
Test level
Solve $4^x - 2^{x + 1} - 15 = 0$ .
Show solution
1. $4^x = (2^x)^2$ and $2^{x + 1} = 2 \cdot 2^x$ . Let $u = 2^x$ : $u^2 - 2u - 15 = 0$ .
2. $(u - 5)(u + 3) = 0$ , so $u = 5$ or $u = -3$ .
3. $2^x = -3$ has no solution, since $2^x > 0$ . Reject it.
4. $2^x = 5$ gives $x = \frac{\log 5}{\log 2} \approx 2.322$ .
5. Check: $4^{2.322} - 2^{3.322} - 15 \approx 25 - 10 - 15 = 0$ .
Answer: $x = \frac{\log 5}{\log 2} \approx 2.322$

Learn It on Martian Lab

Martian Lab teaches this in the Logarithmic & Exponential Equations 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 · Graphing Logarithmic Functions · Exponential and Log Word Problems

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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