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.

## Related Topics

## 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.

- \(\color{blue}{e^x=3 ,x=}\)
- \(\color{blue}{ln(3x-1)=1,x=}\)
- \(\color{blue}{lnx=9,x=}\)
- \(\color{blue}{e^x=9 ,x=}\)
- \(\color{blue}{ln(lnx )=2,x=}\)
- \(\color{blue}{ln(2x+4)=1,x=}\)

- \(\color{blue}{ln(3)}\)
- \(\color{blue}{\frac{e+1}{3}}\)
- \(\color{blue}{e^9}\)
- \(\color{blue}{2ln(3)}\)
- \(\color{blue}{e^{e^2}}\)
- \(\color{blue}{\frac{e-4}{2}}\)