Algebra 2 / Unit 5: Exponential and Logarithmic Functions
5.5 Solving Exponential Equations
Notes
- Same base: if $b^x = b^y$, then $x = y$. Rewrite both sides with a common base.
- Example: $\displaystyle 4^x = 8 \Rightarrow 2^{2x} = 2^3 \Rightarrow x = \frac{3}{2}$.
- Different bases: isolate the power, then take a log of both sides and use the power rule.
- Example: $\displaystyle 3^x = 20 \Rightarrow x = \frac{\ln 20}{\ln 3} \approx 2.727$.
- For $e$, use $\ln$: $e^{x} = 7 \Rightarrow x = \ln 7$.
Practice
Solve $2^{x + 1} = 32$.
$32 = 2^5$, so $x + 1 = 5$.
Solve $9^x = 27$.
$3^{2x} = 3^3$, so $2x = 3$.
Solve $5^x = 40$. Round to three decimals.
$\displaystyle x = \frac{\ln 40}{\ln 5} \approx \frac{3.689}{1.609} \approx 2.292$.
Ready for more? Take the Unit 5 test.