# 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 $$ y = log_ (sqrt (2)) (x) $$.

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