Product Rule: $\log_b(xy)=\log_b(x)+\log_b(y)$
Quotient Rule: $\log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y)$
Power Rule: $\log_b(x^m)=m \log_b(x)$
Using Properties of Logarithms
Examples: Use the properties of logarithms to express the logarithm in terms of logarithms of simpler expressions.
$\ln \left(\frac{w^{7} \sqrt{\gamma}}{h^{8}}\right)$
$\log \left( \frac{11 \xi -12}{2 \xi +4}\right)$
$\ln \frac{(13 \rho - 8)^{-2} (8 \mu + 11)^{4}}{(11 \tau - 15)^{3}}$
$\log \sqrt[3]{6 \zeta +9}$
Dire Warning
$\log_b(xy)$ is NOT equal to $\log_b(x)\log_b(y)$
$\log_b(x+y)$ is NOT equal to $\log_b(x)+\log_b(y)$
$\log_b\left(\frac{x}{y}\right)$ is NOT equal to $\frac{\log_b(x)}{\log_b(y)}$
$\log_b(x-y)$ is NOT equal to $\log_b(x)-\log_b(y)$
Using Properties of Logarithms
Examples: Combine the logarithmic terms into a single logarithmic expression with a coefficient of 1.
$2 \ln (\psi)+\frac{1}{4} \ln (s)$
$6 \log (a)+2\log (p)-8 \log (r)$
$8 \ln (x)-\frac{1}{2}\ln (y)+7 \ln (w)$
$\frac{1}{3} \log (n)-\frac{1}{4} \log (j)$
An Exceedingly Useful Property of Logarithms: The change of base formula. $$\log_b(x)=\frac{\log_a(x)}{\log_a(b)}.$$ Great News! We may now use our calculator to evaluate logarithms of any base!
Example: $\log_5 17$
Example: $\log_{\frac{3}{2}}9$
Bonus Problem: Solve the equation $$3^x=7.$$