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