Logarithms

Related notes: Exponents

Intro

Logs are used to find the power
Example:
$a = b^{c}$
$\log_{b} a = c$

Example

$10^{3} = 1000$
To logarithm:
$3 = \log_{10} 1000$

For common logarithms

For $\log_{b} a = c$

Name	Formula
Product	$$\log_{b}(xy)=\log_{b}{x}+\log_{b}{y}$$
Quotient	$$\log_{b}\left( \frac{x}{y} \right)=\log_{b}x - \log_{b}y$$
Power	$$\log_{b}(x^n)=n\log_{b}x$$
Change of base	$$\log_{b}x=\frac{\log_{c}x}{\log_{c}b}$$
Change of base - Alt Swap base	$$\log_{b}x = \frac{1}{\log_{x}b}$$
Change of base - Alt Combine base	$$\log_{b}x \times \log_{c}b=\log_{c}x$$
Zero	$$\log_{b}1 = 0$$
Identity	$$\log_{b}b=1$$
Equality	$$\log_{b}x=\log_{b}y \implies x=y$$
Inverse	$$b^{\log_{b}x}=x\log_{b}b^x=x$$
Reciprocal	$$\log_{b}{\left( \frac{1}{x} \right)}=-\log_{b}x$$

When calculator doesn't support setting the base

$\log_{a} b = \frac{\lg a}{\lg b}$

$\log_{a} x^{r} = r \log_{a} x$ for any real number r

Proof

Let $m = \log_{a} x$ , so $x = a^{m}$

\begin{aligned} x = a^{m} \\ Make both sides have exponents of r \\ x^{r} & = (a^{m})^{r} \\ = a^{m r} \end{aligned}