Precalculus / Unit 3: Exponential and Logarithmic Functions
3.4 Solving Exponential Equations
Notes
- If the bases can match, set exponents equal: $2^{x + 1} = 32 \Rightarrow x + 1 = 5 \Rightarrow x = 4$.
- Otherwise take a log of both sides and use the power rule: $\displaystyle 5^x = 20 \Rightarrow x = \frac{\ln 20}{\ln 5}$.
- For base $e$, use $\ln$: $e^{2x} = 7 \Rightarrow 2x = \ln 7$.
- Isolate the exponential term first.
Practice
Solve $2^{x + 1} = 32$.
$32 = 2^5$, so $x + 1 = 5$.
Solve $9^x = 3^{x + 4}$.
$3^{2x} = 3^{x + 4}$, so $2x = x + 4$.
Solve $e^{2x} = 7$.
Take $\ln$: $2x = \ln 7$.
Ready for more? Take the Unit 3 test.