Logarithm

tags: Mathematics

All Mathematics Formula by Abhyas here

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Definition

Logarithm of any number

$N$ to the base

$a$ is the power to which the base

$a$ must be raised to obtain

$N$

$$\begin{gathered} {\log }_{a}N=x\iff a^x=N \\ a>0,a\ne 1,N>0 \end{gathered}$$

${\log }_{10}a$ is sometimes written as

$\log a$ and is called Common logarithm.

${\log }_{e}a$ 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

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

Looking at this graph it it evident that

$N$ is always positive

$\mathrm{\forall }a>0$.

Fundamental logarithmic Identity

1. $a^{{\log }_{a}N}=N$

Fundamental Laws of Logarithms

1. ${\log }_{a}m+{\log }_{a}n={\log }_{a}mn$
2. ${\log }_{a}m-{\log }_{a}n={\log }_a\frac{m}{n}$
3. ${\log }_a{m}^n=n{\log }_{a}m$
4. Base Change formula
   
   ${\log }_{a}m=\frac{{\log }_{b}m}{{\log }_{b}a}$
5. Interchange
   
   ${\log }_{a}b=\frac{1}{{\log }_{b}a}$
6. ${\log }_{a^{\beta }}{m}^{\alpha }=\frac{\alpha }{\beta }{\log }_{a}m$
7. ${\log }_{b}a\times {\log }_{a}b=1$
8. $a^{{\log }_{c}b}=b^{{\log }_{c}a}$
9. $a^x=e^{x\ln a}$
10. ${\log }_{10}^{2}x=\ne {\log }_{10}{x}^2$

Graph of logarithmic functions

Graph 1

1. $y={\log }_{2}x$ 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
   $(1,0)$
   
   ${\log }_{a}1=0$
2. No graph for -ve
   $x$
3. For
   $a>1$
   1. ${\log }_{a}0=-\mathrm{\infty }$
   2. Increasing graph
4. For
   $0<a<1$
   1. $lo{g}_{a}0=\mathrm{\infty }$
   2. Decreasing graph

Graph 2

1. $y={\log }_{10}x$ in Red
2. $y={\log }_{e}x$ in Blue
3. $y={\log }_{2}x$ in Orange

Note:

$2<e<10$

Graph 3

1. $y={\log }_{\frac{1}{10}}x$ in Red
2. $y={\log }_{\frac{1}{e}}x$ in Blue
3. $y={\log }_{\frac{1}{2}}x$ 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

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.