Solving Exponential and Logarithmic Equations

Exponential Equations

The variable is in the exponent:

$2^x = 8 \qquad 3^x = 20 \qquad e^x = 5$

If both sides have the same base, match the exponents.

Example: Solve $2^x = 8$

Rewrite 8 as a power of 2:

$2^x = 2^3$

Same base, so exponents are equal:

$x = 3$

Example: Solve $3^{2x} = 81$

Rewrite 81 as a power of 3:

$3^{2x} = 3^4$

Match exponents:

$2x = 4 \quad \Rightarrow \quad x = 2$

When you can’t match bases, take the log of both sides.

Key property: $\log(a^n) = n \cdot \log(a)$

Example: Solve $2^x = 10$

Can’t write 10 as a power of 2. Take log of both sides:

\begin{aligned} \log(2^x) &= \log(10) \\ x \cdot \log(2) &= 1 \\ x &= \frac{1}{\log(2)} \\ x &\approx 3.32 \end{aligned}

Example: Solve $e^x = 7$

Take ln of both sides (natural log pairs with base $e$ ):

\begin{aligned} \ln(e^x) &= \ln(7) \\ x &= \ln(7) \\ x &\approx 1.95 \end{aligned}

Example: Solve $5^{x+1} = 30$

\begin{aligned} \log(5^{x+1}) &= \log(30) \\ (x+1) \cdot \log(5) &= \log(30) \\ x + 1 &= \frac{\log(30)}{\log(5)} \\ x &= \frac{\log(30)}{\log(5)} - 1 \\ x &\approx 1.11 \end{aligned}

The variable is inside the log:

$\log(x) = 2 \qquad \ln(x + 1) = 3$

Strategy: Convert to exponential form.

Example: Solve $\log(x) = 2$

Convert to exponential form:

$10^2 = x$

$x = 100$

Example: Solve $\ln(x) = 4$

Convert to exponential form:

$e^4 = x$

$x \approx 54.6$

Example: Solve $\log_2(x - 1) = 5$

Convert to exponential form:

\begin{aligned} 2^5 &= x - 1 \\ 32 &= x - 1 \\ x &= 33 \end{aligned}