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

  1. 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.
  2. The natural logarithm of x is generally written as ln x, or \(\log_{e}{x}\).

Example 1:

Solve the equation for \(x\): \(e^x=3\)

Answer:

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

Answer:

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 \(x\): \(e^x=5\)

Answer:

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 \(x\): \(ln(5x-1)=1\)

Answer:

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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