How to Solve Natural Logarithms Problems

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.

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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: \(x=ln(3) \)

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. \(ln(2x-1)=1\)

Answers

  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. \(x=\frac{e+1}{2}\)

Related to "How to Solve Natural Logarithms Problems"

How to Solve Natural Logarithms
How to Solve Natural Logarithms

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