Logarithms — Examples & Problems

Definition

\log_b x=y\iff b^y=x

Domain

x>0,\qquad b>0,\qquad b\ne1

Range

(-\infty,+\infty)

If $b>1$ , then $y=log_b x$ is increasing.

If $0<b<1$ , then $y=log_b x$ is decreasing.

Properties

\begin{aligned} 1.\quad & \log_b1=0\\[10pt] 2.\quad & \log_b b=1\\[10pt] 3.\quad & \log_b(b^m)=m\\[10pt] 4.\quad & b^{\log_b a}=a\\[10pt] 5.\quad & \log_b a+\log_b c=\log_b(ac)\\[10pt] 6.\quad & \log_b a-\log_b c=\log_b\left(\frac{a}{c}\right)\\[10pt] 7.\quad & \log_b(a^n)=n\log_b a\\[10pt] 8.\quad & \log_{b^m}a=\frac{1}{m}\log_b a\\[10pt] 9.\quad & \log_b\left(\frac{1}{a}\right)=-\log_b a\\[10pt] 10.\quad & \log_b a\cdot\log_a c=\log_b c\\[10pt] 11.\quad & \log_b a=\frac{\log_c a}{\log_c b}\\[10pt] 12.\quad & \log_b a=\frac{\ln a}{\ln b}, \qquad \text{where }\ln x=\log_e x,\ e=2.71828\ldots\\[10pt] 13.\quad & \log_b a=\frac{1}{\log_a b}\\[10pt] 14.\quad & b^{(\log_a x)(\log_b a)}=x\\[10pt] 15.\quad & \log_b x=\log_b y\iff x=y \end{aligned}