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

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.

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

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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 \(2.17\). 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. \(\color{blue}{ln⁡(2x+4)=1,x=}\)
This image has an empty alt attribute; its file name is answers.png
  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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