Logarithm properties review

Review the logarithm properties and how to apply them to solve problems.

What are the logarithm properties?

Product rule$\large\log_b(MN)=\log_b(M)+\log_b(N)$
Quotient rule$\large\log_b\left(\frac{M}{N}\right)=\log_b(M)-\log_b(N)$
Power rule$\large\log_b(M^p)=p\log_b(M)$
Change of base rule$\large\log_b(M)=\dfrac{\log_a(M)}{\log_a(b)}$

Rewriting expressions with the properties

We can use the logarithm properties to rewrite logarithmic expressions in equivalent forms.
For example, we can use the product rule to rewrite $\log(2x)$ as $\log(2)+\log(x)$. Because the resulting expression is longer, we call this an expansion.
In another example, we can use the change of base rule to rewrite $\dfrac{\ln(x)}{\ln(2)}$ as $\log_2(x)$. Because the resulting expression is shorter, we call this a compression.
Want to try more problems like this? Check out this exercise.

Evaluating logarithms with calculator

Calculators usually only calculate $\log$ (which is log base $10$) and $\ln$ (which is log base $e$).
Suppose, for example, we want to evaluate $\log_2(7)$. We can use the change of base rule to rewrite that logarithm as $\dfrac{\ln(7)}{\ln(2)}$ and then evaluate in the calculator:
\begin{aligned} \log_2(7)&=\dfrac{\ln(7)}{\ln(2)} \\\\ &\approx 2{,}807 \end{aligned}
Want to try more problems like this? Check out this exercise.