# How to Solve Natural Logarithms

Logarithms that have Base e (natural logarithms) are important in mathematics and some scientific applications. This blog post explains the applications of natural logarithms with examples.

## Definition of Natural Logarithms

A natural logarithm is a logarithm that has a special base of the mathematical constant $$e$$, which is an irrational number approximately equal to 2.71. The natural logarithm of $$x$$ is generally written as $$ln \ x$$, or $$log_{e}{x}$$.

## Examples

### Natural Logarithms – Example 1:

Solve this equation for $$x: e^x=6$$

Solution:

If $$f(x)=g(x)$$,then: $$ln(f(x))=ln(g(x))→ln(e^x)=ln(6)$$
Use log rule: $$log_{a}{x^b }=b \ log_{a⁡}{x}→ ln(e^x)=x \ ln(e)→x \ ln(e)=ln(6)$$
$$ln(e)=1$$, then: $$x=ln(6)$$

### Natural Logarithms – Example 2:

Solve this equation for $$x: ln(4x-2)=1$$

Solution:

Use log rule: $$a=log{_b⁡}{b^a}→1=ln⁡(e^1 )=ln⁡(e)→ln⁡(4x-2)=ln⁡(e)$$ When the logs have the same base: $$log_{b}{(f(x))}=log_{b }{(g(x))}→f(x)=g(x)$$
$$ln(4x-2)=ln(e)$$, then: $$4x-2=e→x=\frac{e+2}{4}$$

### Natural Logarithms – Example 3:

Solve this equation for $$x: ln(3x-4)=1$$

Solution:

Use log rule: $$a=log_{b⁡}{(b^a)}→1=ln⁡(e^1 )=ln⁡(e)→ln⁡(3x-4)=ln⁡(e)$$
When the logs have the same base: $$log_{b⁡}{(f(x))}=log_{b}{ (g(x))}→f(x)=g(x)$$
$$ln(3x-4)=ln(e)$$, then: $$3x-4=e→x=\frac{e+4}{3}$$

### Natural Logarithms – Example 4:

Solve this equation for $$x: ln(5x+8)=1$$

Solution:

Use log rule: $$a=log_{b⁡}{(b^a)}→1=ln⁡(e^1 )=ln⁡(e)→ln⁡(5x+8)=ln⁡(e)$$
When the logs have the same base: $$log_{b⁡}{(f(x))}=log_{b}{ (g(x))}→f(x)=g(x)$$
$$ln(5x+8)=ln(e)$$, then: $$5x+8=e→x=\frac{e-8}{5}$$

## Exercises for Natural Logarithms

### Find the value of $$x$$ in each equation.

1. $$\color{blue}{e^x=3 ,x=}$$
2. $$\color{blue}{ln⁡(3x-1)=1,x=}$$
3. $$\color{blue}{ln⁡x=9,x=}$$
4. $$\color{blue}{e^x=9 ,x=}$$
5. $$\color{blue}{ln⁡(ln⁡x )=2,x=}$$
6. $$\color{blue}{ln⁡(2x+4)=1,x=}$$
1. $$\color{blue}{ln⁡(3)}$$
2. $$\color{blue}{\frac{e+1}{3}}$$
3. $$\color{blue}{e^9}$$
4. $$\color{blue}{2ln(3)}$$
5. $$\color{blue}{e^{e^2}}$$
6. $$\color{blue}{\frac{e-4}{2}}$$

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