# What is log XY

### Logarithm and exponentiation

As a reminder: The logarithm of \$\$ y \$\$ to the base \$\$ b \$\$ is the number \$\$ x \$\$ with which one must raise \$\$ b \$\$ to the power to get \$\$ y \$\$.

Examples:
a) \$\$ log_2 (8) = 3 \$\$, since \$\$ 2 ^ 3 = 8 \$\$.

b) \$\$ log_2 (32) = 4 \$\$, since \$\$ 2 ^ 4 = 3 \$\$

c) \$\$ log_9 (1/81) = log_9 (1 / (9 ^ 2)) \$\$\$\$ = log_9 (9 ^ -2) = - 2 \$\$,
since \$\$ 9 ^ -2 = 1/81 \$\$

\$\$ b ^ x = y \$\$ means the same as \$\$ log_b (y) = x \$\$.
(It applies \$\$ b> 0 \$\$, \$\$ y> 0 \$\$ and \$\$ b ≠ 1 \$\$)

You can reverse the high - \$\$ x \$\$ - with the logarithm.

Calculating high - \$\$ x \$\$ - and taking the logarithm are therefore reverse operations.

\$\$ log_b (b ^ x) = x \$\$ and \$\$ b ^ (log_b x) = x \$\$.

##### Logarithmic Laws:

For logarithms based on \$\$ b \$\$ with \$\$ b ≠ 1 \$\$ and \$\$ b> 0 \$\$ and for positive real numbers \$\$ u \$\$ and \$\$ v \$\$ as well as a real number \$\$ r \$\$ applies:

\$\$ log_b (u * v) = log_b (u) + log_b (v) \$\$

\$\$ log_b (u / v) = log_b (u) -log_b (v) \$\$

\$\$ log_b (u ^ r) = r * log_b (u) \$\$

### Logarithmic function of the exponential function

You can transfer this reverse principle to functions.

The exponential function \$\$ y = b ^ x \$\$ \$\$ (b ≠ 0) \$\$ and the logarithm function \$\$ y = log_b (x) \$\$ (with \$\$ x> 0 \$\$, \$\$ b> 0 \$\$ , \$\$ b ≠ 1 \$\$) are inverse functions.

The logarithm function is graphically created by mirroring the exponential function on the straight line through the origin \$\$ y = x \$\$.

Example:
\$\$ f (x) = 2 ^ x \$\$ and \$\$ g (x) = log_2 (x) \$\$.

A function with the equation \$\$ y = log_b (x) \$\$ with \$\$ x> 0 \$\$ is called a logarithm function based on \$\$ b \$\$, where \$\$ b> 0 \$\$ and \$\$ b ≠ 1 \$\$.

The logarithm function \$\$ y = log_b (x) \$\$ is the inverse function of the exponential function \$\$ y = b ^ x \$\$. The graphs are mirror images of the straight line from \$\$ y = x \$\$.

The inverse function of \$\$ f \$\$ is also referred to as \$\$ f ^ -1 \$\$.

### Determine the functional equation from a point

You have seen that the graphs of the logarithmic functions \$\$ y = log_b (x) \$\$ only have exactly one point in common. Every other point therefore clearly defines a logarithm function. When you have a point, you can determine the function!

Examples:
Find the logarithm function that passes through the point \$\$ (2 | 2) \$\$. To do this, you insert the coordinates of \$\$ P \$\$ and convert to \$\$ b \$\$:

\$\$ y = log_b (x) \$\$

\$\$ & Leftrightarrow; 2 = log_b (2) \$\$

\$\$ & Leftrightarrow; b ^ 2 = 2 \$\$ \$\$ | sqrt \$\$

\$\$ & Leftrightarrow; b = + - sqrt (2) \$\$

\$\$ - sqrt (2) \$\$ can be excluded as a solution. Therefore the function equation we are looking for is \$\$ y = log_ (sqrt (2)) (x) \$\$.