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disqus: abhyas29
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# Logarithm

###### tags: `Mathematics`

[All Mathematics Formula by Abhyas here](/@abhyas/maths_formula)

Please see [README](https://hackmd.io/@abhyas/maths_formula#README) if this is the first time you are here.

## Definition

Logarithm of any number $N$ to the base $a$ is the power to which the base $a$ must be raised to obtain $N$

$\log_aN=x \Leftrightarrow a^x=N\\
a > 0, a \ne 1, N > 0$

$\log_{10}a$ is sometimes written as $\log a$ and is called Common logarithm.

$\log_ea$ is sometimes written as $\ln a$ and is called natural logarithm.

## Graph of Exponential Functions

Graph in the form $N = a^x$
y-axis is $N$
x-axis is $x$
a is different for each graph
<iframe src="https://www.desmos.com/calculator/jmun0w803b?embed" width="479px" height="479px" style="border: 1px solid #ccc" frameborder=0></iframe>

1. $y = 5^x$ in Blue
1. $y = 2^x$ in Red
1. $y = 0.5^x$ in Orange

Looking at this graph it it evident that $N$ is always positive $\forall a > 0$.

## Fundamental logarithmic Identity

1. $a^{\log_aN}=N$

## Fundamental Laws of Logarithms

1. $\log_am+\log_an=\log_amn$
2. $\log_am-\log_an=\log_a\frac{m}{n}$
3. $\log_am^n=n\log_am$
4. Base Change formula
$\log_am = \frac{\log_bm}{\log_ba}$
5. Interchange
$\log_ab=\frac{1}{\log_ba}$
6. $\log_{a^\beta}m^\alpha=\frac{\alpha}{\beta}\log_am$
7. $\log_ba\times\log_ab=1$
8. $a^{\log_cb}=b^{\log_ca}$
9. $a^x=e^{x\ln{a}}$
10. $\log^2_{10}x=\ne \log_{10}x^2$

## Graph of logarithmic functions

### Graph 1

<iframe src="https://www.desmos.com/calculator/e81l79kmck?embed" width="479px" height="479px" style="border: 1px solid #ccc" frameborder=0></iframe>

1. $y = \log_2x$ in Blue
Here $a>1$
2. $y = \log_{0.3}x$ in Red
Here $0 < a < 1$

#### Observations from graph

1. All $\log$ graphs passes through $(1,0)$
$\log_a1 = 0$
2. No graph for -ve $x$
3. For $a>1$
1. $\log_a0 = -\infty$
2. Increasing graph
4. For $0<a<1$
1. $log_a0=\infty$
2. Decreasing graph

### Graph 2

<iframe src="https://www.desmos.com/calculator/hcmfpux3vg?embed" width="479px" height="479px" style="border: 1px solid #ccc" frameborder=0></iframe>

1. $y = \log_{10}x$ in Red
2. $y = \log_ex$ in Blue
3. $y = \log_2x$ in Orange

Note: $2 < e < 10$

### Graph 3

<iframe src="https://www.desmos.com/calculator/xpyb8kpsq8?embed" width="479px" height="479px" style="border: 1px solid #ccc" frameborder=0></iframe>

1. $y = \log_{\frac{1}{10}}x$ in Red
2. $y = \log_{\frac{1}{e}}x$ in Blue
4. $y = \log_{\frac{1}{2}}x$ in Orange

Note: $\frac{1}{10} < \frac{1}{e} < \frac{1}{2}$

## Logarithmic Equations

### Constant Base

$\log_a{f(x)}=\log_a{g(x)}$

Condition to solve: $f(x)>0$ and $g(x) >0$
To solve: Take anti-log to get $f(x)=g(x)$
Final solution = condition $\cap$ solution

### Function as Base

$\log_{h(x)}{f(x)}=\log_{h(x)}{g(x)}$

Condition to solve: $f(x)>0$ and $g(x) >0$ and $h(x) > 0$ and $h(x) \ne 1$
To solve: Take anti-log to get $f(x)=g(x)$
Final solution = condition $\cap$ solution

## Licensing and Links

[All Mathematics Formula by Abhays here](/Uhf7AR-EQcKrvHaPG9FSXg)

<a rel="license" href="http://creativecommons.org/licenses/by-nc/4.0/"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-nc/4.0/88x31.png" /></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-nc/4.0/">Creative Commons Attribution-NonCommercial 4.0 International License</a>.