Training Exponentials & Logarithms Solving Exponential & Logarithmic Equations
4 / 5

Solving Exponential & Logarithmic Equations

25 min Exponentials & Logarithms

Solving Exponential & Logarithmic Equations

Solving exponential and logarithmic equations brings together everything you have learned in this module. Exponential equations are solved by taking the logarithm of both sides, while logarithmic equations are solved by rewriting them in exponential form.

Many problems require you to apply the properties of logarithms to simplify the equation before solving. This lesson covers both types and emphasizes checking for extraneous solutions, which can arise when the argument of a logarithm becomes negative or zero.

Type 1: Same-Base Exponential Equations

Rule

If $b^m = b^n$ then $m = n$ (one-to-one property).

Example 1

Solve $2^{3x-1} = 16$.

Write $16 = 2^4$: $\;2^{3x-1} = 2^4$.

$3x - 1 = 4 \implies x = \dfrac{5}{3}$.

Type 2: Using Logarithms

Strategy

If the bases cannot be made equal, take the logarithm of both sides.

Example 2

Solve $5^x = 20$.

$\ln(5^x) = \ln(20)$

$x \ln(5) = \ln(20)$

$$x = \frac{\ln(20)}{\ln(5)} \approx \frac{2.996}{1.609} \approx 1.861$$

Example 3

Solve $3^{2x+1} = 7^{x-2}$.

Take $\ln$ of both sides:

$(2x+1)\ln 3 = (x-2)\ln 7$

$2x\ln 3 + \ln 3 = x\ln 7 - 2\ln 7$

$x(2\ln 3 - \ln 7) = -2\ln 7 - \ln 3$

$$x = \frac{-2\ln 7 - \ln 3}{2\ln 3 - \ln 7} \approx \frac{-4.990}{0.249} \approx -20.04$$

Type 3: Logarithmic Equations

Example 4

Solve $\log_2(x-3) = 4$.

Convert to exponential: $x - 3 = 2^4 = 16 \implies x = 19$.

Check: $\log_2(16) = 4$ ✓

Example 5

Solve $\log(x) + \log(x+3) = 1$.

Condense: $\log[x(x+3)] = 1$.

$x(x+3) = 10^1 = 10$.

$x^2 + 3x - 10 = 0 \implies (x+5)(x-2) = 0$.

$x = -5$ or $x = 2$.

Check $x=-5$: $\log(-5)$ is undefined ✗

Check $x=2$: $\log(2) + \log(5) = \log(10) = 1$ ✓

Solution: $x = 2$.

Example 6

Solve $\ln(2x+1) = 3$.

$2x + 1 = e^3 \implies x = \dfrac{e^3 - 1}{2} \approx 9.543$

Practice Problems

1. Solve $4^x = 64$
2. Solve $3^x = 50$
3. Solve $2^{x+3} = 32$
4. Solve $e^{2x} = 15$
5. Solve $\log_3(x) = 4$
6. Solve $\log(x) + \log(x-9) = 1$
7. Solve $5^{x-1} = 125$
8. Solve $\ln(x) = 2$
9. Solve $7^{2x} = 3^{x+5}$
10. Solve $\log_2(x) + \log_2(x-6) = 4$
11. Solve $10^{x-1} = 500$
12. How long to double $\$1{,}000$ at 6% compounded continuously? ($2000 = 1000e^{0.06t}$)
13. Solve $\log_5(3x+1) = 2$
14. Solve $2\ln(x) = 8$
15. Solve $9^x = 27$
Show Answer Key

1. $x = 3$

2. $x = \frac{\ln 50}{\ln 3} \approx 3.561$

3. $x = 2$  ($2^5 = 32$)

4. $x = \frac{\ln 15}{2} \approx 1.354$

5. $x = 81$

6. $x = 10$  ($x = -1$ is extraneous)

7. $x = 4$  ($5^3 = 125$)

8. $x = e^2 \approx 7.389$

9. $x = \frac{5\ln 3}{2\ln 7 - \ln 3} \approx 2.087$

10. $x = 8$  ($x = -2$ is extraneous)

11. $x = 1 + \log(500) \approx 3.699$

12. $t = \frac{\ln 2}{0.06} \approx 11.55$ years

13. $x = 8$

14. $x = e^4 \approx 54.598$

15. $x = \frac{3}{2}$  ($9 = 3^2$, $27 = 3^3$, so $3^{2x} = 3^3$)

Properties of Logarithms Placement Test Practice — Exponentials & Logs