$(x-4)(x+2) = 0$

Possible solutions: $x=4$ or $x=-2$

Check Domain: Argument of log must be positive.

When the variable is in the exponent, take the logarithm of both sides.
Example: Solve $3^x = 7$
- $\log(3^x) = \log(7)$ (can use any base, $\ln$ or $\log_{10}$ are common)
- $x \log(3) = \log(7)$ (Power Rule)
- $x = \frac{\log(7)}{\log(3)}$
- $x \approx \frac{0.845}{0.477} \approx 1.771$
Example: Solve $e^{2x+1} = 5$
- $\ln(e^{2x+1}) = \ln(5)$ (use natural log for base $e$)
- $2x+1 = \ln(5)$ (since $\ln(e^k)=k$)
- $2x = \ln(5) - 1$
- $x = \frac{\ln(5) - 1}{2}$
- $x \approx \frac{1.609 - 1}{2} \approx \frac{0.609}{2} \approx 0.3045$

General Form: $y = \log_b(x)$
Domain: $(0, \infty)$ (i.e., $x > 0$)
Range: $(-\infty, \infty)$
x-intercept: $(1, 0)$ (because $\log_b(1)=0$)
Vertical Asymptote: $x=0$ (the y-axis)
Shape:
- If $b > 1$: Increasing function, concave down.
- If $0 < b < 1$: Decreasing function, concave up.
Relationship to Exponential: The graph of $y = \log_b(x)$ is a reflection of $y = b^x$ across the line $y=x$.