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

What are the logarithm properties?

Product rulelogb(MN)=logb(M)+logb(N)\large\log_b(MN)=\log_b(M)+\log_b(N)
Quotient rulelogb(MN)=logb(M)logb(N)\large\log_b\left(\frac{M}{N}\right)=\log_b(M)-\log_b(N)
Power rulelogb(Mp)=plogb(M)\large\log_b(M^p)=p\log_b(M)
Change of base rulelogb(M)=loga(M)loga(b)\large\log_b(M)=\dfrac{\log_a(M)}{\log_a(b)}
Want to learn more about logarithm properties? Check out this video.

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)\log(2x) as log(2)+log(x)\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 ln(x)ln(2)\dfrac{\ln(x)}{\ln(2)} as log2(x)\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\log (which is log base 1010) and ln\ln (which is log base ee).
Suppose, for example, we want to evaluate log2(7)\log_2(7). We can use the change of base rule to rewrite that logarithm as ln(7)ln(2)\dfrac{\ln(7)}{\ln(2)} and then evaluate in the calculator:
log2(7)=ln(7)ln(2)2,807\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.
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