Integrated Math 3 / Unit 3: Exponential and Logarithmic Functions
3.4 Solving Exponential Equations
Notes
- If the bases can match, set the exponents equal: $\displaystyle 9^x = 27 \Rightarrow 3^{2x} = 3^3 \Rightarrow x = \frac{3}{2}$.
- Otherwise isolate the power, then take a log of both sides: $\displaystyle 3^x = 20 \Rightarrow x = \frac{\ln 20}{\ln 3} \approx 2.73$.
- Isolate first: $5 \cdot 2^x = 80 \Rightarrow 2^x = 16 \Rightarrow x = 4$.
Practice
Solve $4^{x + 1} = 64$.
$4^{x + 1} = 4^3$, so $x + 1 = 3$.
Solve $5 \cdot 2^x = 80$.
Divide by 5: $2^x = 16 = 2^4$.
Which is the best estimate of the solution of $3^x = 20$?
$\displaystyle x = \frac{\ln 20}{\ln 3} \approx \frac{2.996}{1.099} \approx 2.73$. Since $3^2 = 9$ and $3^3 = 27$, the answer is between 2 and 3.
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