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

Related Topics

• How to Solve Logarithmic Equations
• How to Evaluate Logarithms
• Properties of 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. \(\color{blue}{ln(2x-1)=1}\)

Answers

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