Logarithm with arbitrary base

Summary

• For any given base $b>0$ and any given real number $c>0$ the equation

$$b^a=c$$

• has a unique solution $a$ .
• The solution $a$ is denoted by ${log}_b c$ and pronounced the logarithm of $c$ to base $b$ .
• For example, ${log}_2 8={log}_2 2^3=3$ and ${log}_4 0.25={log}_4 4^{-1}=-1$ .
• If $c≤0$ then ${log}_b c$ is not defined.

Logarithmic identities and laws

• ${log}_b b^a=a$ for all $a$
• $b^{{log}_b c}=c$ for $c>0$

Logarithm of products, ratios and powers

• ${log}_b cd={log}_b c+{log}_b d$
• ${log}_b \frac{c}{d}={log}_b c-{log}_b d$
• ${log}_b c^d=d{log}_b c$
• ${log}_b c= \frac{ln c}{ln b}= \frac{log c}{log b}$
• Given $b^a=c$ where $c>0$ is given,
• You get the base $b$ from exponent $a$ using a radical, $b=\sqrt[a]{c}$
• You get the exponent $a$ from the base $b$ using a logarithm, $a={log}_b c$

Lecture coming soon