Natural Logarithms

Natural Logarithms

Do you want to know how to solve Natural Logarithms? you can do it in two easy steps.

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 \(\log_{e}{x}\).

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: \(x=ln(3) \)

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: \(ln(2x-1)=ln(e), then: 2x-1=e→x=\frac{e+1}{2}\)

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: \(x=ln(5) \)

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

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

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