# How to Solve Natural Logarithms Problems

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

## 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.

1. $$e^x=3$$
2. $$e^x=4$$
3. $$e^x=8$$
4. $$ln x=6$$
5. $$ln (ln x)=5$$
6. $$e^x=9$$
7. $$ln⁡(2x+5)=4$$
8. $$ln(2x-1)=1$$

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

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