# What is the graph of y logx

### Logarithm and exponentiation

As a reminder: The logarithm of $$ y $$ to the base $$ b $$ is the number $$ x $$ with which one has to 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 $$ log_b (y) = x $$.

(The following 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 $$ y = log_ (sqrt (2)) (x) $$.

- What is a comparative market analysis
- Why do arcade games cost so much
- What is the name of Santa Claus in Portuguese
- How do you see anonymous questions
- How would you cast for a movie
- What are the drugs for watery semen
- What's your favorite angel cake recipe
- What animals are used for black magic
- Why will Bitcoin stop producing coins at some point?

- Which industry does blockchain belong to?
- Hillary Clinton is lying 1
- What is Libaio on Linux
- What is fishing in the oil industry
- If it's not perpendicular, then what is it
- What are the uses of hydrogen bonding
- What is the etymology of pusillanimous
- Working fans in hot weather
- Is it possible to be under stress?
- What is Mumbai's GDP Growth Rate
- Is it an earthworm or a parasite
- How big is the market for industrial communication
- Nostalgia definitely makes us sad
- Is the GMAT sentence correction useful for CAT
- Why is PKM Personal Knowledge Management important
- Which event will make climate change undeniable?
- How can you see Apple WWDC 2019
- Why don't you feel inferior?
- How strong is a magnitude 6 earthquake
- A rattlesnake has a backbone
- Who was the first humanoid robot
- Is Kanye West right now
- How do you drain your water pipes
- What percentage of life is just nonsense