In this blog post, you will learn more about Natural Logarithms and how to solve problems related to natural logarithms.

## Related Topics

## Step by step guide to solve 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 \(\log_{e}{x}\).

### Natural Logarithms – Example 1:

Solve the equation for \(x\): \(e^x=3\)

**Solution:**

If \(f(x)=g(x),then: ln(f(x))=ln(g(x))→ln(e^x)=ln(3) \)

Use log rule: \(\log_{a}{x^b}=b \log_{a}{x}\), then: \(ln(e^x)=x ln(e)→xln(e)=ln(3) \)

\(ln(e)=1\), then: \(x=ln(3) \)

### Natural Logarithms – Example 2:

Solve equation for \(x\): \(ln(2x-1)=1\)

**Solution:**

Use log rule: \(a=\log_{b}{b^a}\), then: \(1=ln(e^1 )=ln(e)→ln(2x-1)=ln(e)\)

When the logs have the same base: \(\log_{b}{f(x)}=\log_{b}{g(x)}\), then: \(f(x)=g(x)\)

then: \(ln(2x-1)=ln(e), then: 2x-1=e→x=\frac{e+1}{2}\)

### Natural Logarithms – Example 3:

Solve the equation for \(x\): \(e^x=5\)

**Solution:**

If \(f(x)=g(x),then: ln(f(x))=ln(g(x))→ln(e^x)=ln(5) \)

Use log rule: \(\log_{a}{x^b}=b \log_{a}{x}\), then: \(ln(e^x)=x ln(e)→xln(e)=ln(5) \)

\(ln(e)=1\), then: \(x=ln(5) \)

### Natural Logarithms – Example 4:

Solve equation for \(x\): \(ln(5x-1)=1\)

**Solution:**

Use log rule: \(a=\log_{b}{b^a}\), then: \(1=ln(e^1 )=ln(e)→ln(5x-1)=ln(e)\)

When the logs have the same base: \(\log_{b}{f(x)}=\log_{b}{g(x)}\), then: \(f(x)=g(x)\)

then: \(ln(5x-1)=ln(e), then: 5x-1=e→x=\frac{e+1}{5}\)

## Exercises to practice Natural Logarithms

### Solve each equation for x.

- \(e^x=3\)
- \(e^x=4\)
- \(e^x=8\)
- \(ln x=6\)
- \(ln (ln x)=5\)
- \(e^x=9 \)
- \(ln(2x+5)=4\)
- \(ln(2x-1)=1\)

## Answers

- \(x=ln 3\)
- \(x=ln 4,x=2ln(2)\)
- \(x=ln 8,x=3ln(2)\)
- \(x=e^6 \)
- \(x=e^{e^5}\)
- \(x=ln 9,x=2ln(3)\)
- \(x=\frac{e^4-5}{2}\)
- \(x=\frac{e+1}{2}\)