# How to Solve Natural Logarithms Problems? (+FREE Worksheet!)

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

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

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