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.
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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: \(x=ln(3) \)
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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
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Solve each equation for \(x\).
- \(\color{blue}{e^x=3}\)
- \(\color{blue}{e^x=4}\)
- \(\color{blue}{e^x=8}\)
- \(\color{blue}{ln x=6}\)
- \(\color{blue}{ln (ln x)=5}\)
- \(\color{blue}{e^x=9}\)
- \(\color{blue}{ln(2x+5)=4}\)
- \(\color{blue}{ln(2x-1)=1}\)
Answers
- \(\color{blue}{x=ln 3}\)
- \(\color{blue}{x=ln 4,x=2ln(2)}\)
- \(\color{blue}{x=ln 8,x=3ln(2)}\)
- \(\color{blue}{x=e^6}\)
- \(\color{blue}{x=e^{e^5}}\)
- \(\color{blue}{x=ln 9,x=2ln(3)}\)
- \(\color{blue}{x=\frac{e^4-5}{2}}\)
- \(\color{blue}{x=\frac{e+1}{2}}\)
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