# Functions with Base b

## Key Questions

• Log/Exp Inverse Properties

${b}^{{\log}_{b} x} = x$

${\log}_{b} {b}^{x} = x$

Other Log Properties

${\log}_{b} \left(x \cdot y\right) = {\log}_{b} x + {\log}_{b} y$

${\log}_{b} \left(\frac{x}{y}\right) = {\log}_{b} x - {\log}_{b} y$

${\log}_{b} {x}^{r} = r {\log}_{b} x$

I hope that this was helpful.

${\log}_{a} \left({m}^{n}\right) = n {\log}_{a} \left(m\right)$

#### Explanation:

Consider the logarithmic number ${\log}_{a} \left(m\right) = x$:

${\log}_{a} \left(m\right) = x$

Using the laws of logarithms:

$\implies m = {a}^{x}$

Let's raise both sides of the equation to $n$th power:

$\implies {m}^{n} = {\left({a}^{x}\right)}^{n}$

Using the laws of exponents:

$\implies {m}^{n} = {a}^{x n}$

Let's separate $x n$ from $a$:

$\implies {\log}_{a} \left({m}^{n}\right) = x n$

Now, we know that ${\log}_{a} \left(m\right) = x$.

Let's substitute this in for $x$:

$\implies {\log}_{a} \left({m}^{n}\right) = n {\log}_{a} \left(m\right)$

The reflection of the exponential function on the axis $y = x$

#### Explanation:

Logarithms are the inverse of an exponential function, so for $y = {a}^{x}$, the log function would be $y = {\log}_{a} x$.

So, the log function tell you what power $a$ has to be raised to, to get $x$.

Graph of $\ln x$:
graph{ln(x) [-10, 10, -5, 5]}

Graph of ${e}^{x}$:
graph{e^x [-10, 10, -5, 5]}