What is Natural Logarithm

What is Natural Logarithm

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ln(x) - Natural Logarithm

The natural logarithm of a number is its logarithm to the base, \(e\).

What is natural logarithm?

If \(e^y \ = \ x\), the base \(e\) logarithm of \(x\) is \(ln(x) \ = \ log_e \ x \ = \ y\)
The \(e\) constant also called Euler's number, \(e \ = \ 2.71828183\)

Inverse Function of Exponential Function - ln

The exponential function \(e^x\) (\(f(x) \ = \ e^x\)) is the inverse of the natural logarithm function, \(ln(x)\) \((f^{-1}(x) \ = \ ln(x))\).
For \(x \ > \ 0\),

  • \(f(f^{-1}(x)) \ = \ e^{ln(x)} \ = \ x\)
  • \(f^{-1}(f(x)) \ = \ ln(e^x) \ = \ x\)

The Rules and Properties of the Natural Logarithm

Product Rule

  • \(ln \ (m)(n) \ = \ ln \ (m) \ + \ ln \ (n)\)
  • The natural \(log\) of \(x\) times \(y\) is the sum of the log of \(x\) and \(y\).

Example: \(ln \ (7 \times 5) \ = \ ln \ 7 \ + \ ln \ 5\)

Quotient Rule

  • \(ln \ (\frac{m}{n}) \ = \ ln \ (m) \ - \ ln \ (n)\)
  • The natural log of dividing \(m\) by \(n\) is the difference between the natural \(log\) of \(m\) and \(n\).

Example: \(ln \ (\frac{9}{11}) \ = \ ln \ (9) \ - \ ln \ (11)\)

Reciprocal Rule

  • \(ln \ (\frac{1}{n}) \ = \ -ln \ (n)\)
  • The opposite of the \(ln\) of \(x\) is the natural \(log\) of the reciprocal of \(x\).

Example: \(ln \ (\frac{1}{10}) \ = \ -ln \ (10)\)

Power Rule

  • \(ln \ (x^y) \ = \ yln \ (x)\)
  • The natural log of \(x\) to the power of \(y\) is \(y\) multiplied by the natural \(log\) of \(x\).

Example: \(ln \ 5^3 \ = \ 3ln \ 5\)

ln of 0

  • \(ln \ 0\) is undefined.
  • The natural logarithm of \(0\) is undefined.

ln of 1

  • \(ln \ 1 \ = \ 0\)
  • The natural logarithm of \(1\) is \(0\).

ln of e

  • \(ln \ (e) \ = \ 1\)
  • "One is the natural logarithm of \(e\)."

ln of e raised to the x power

  • \(ln \ e^x \ = \ x\)
  • The natural logarithm of \(e^x\) is \(x\). 

e raised to the ln power

  • \(e^{ln \ x} \ = \ x\)
  • \(e\) raised to the \(ln\) power is equal to \(x\).

Free printable Worksheets

Exercises for Natural Logarithms

1) Evaluate without using a calculator: \(3 \ ln \ e\)

2) Evaluate without using a calculator: \(-3 \ ln \ e^5\)

3) Evaluate without using a calculator: \(9 \ ln \ \frac{1}{e}\)

4) Evaluate without using a calculator: \(6 \ ln \ \frac{1}{e^5}\)

5) Evaluate without using a calculator: \(e^{ln \ \frac{1}{e}}\)

6) Solve for \(x\): \(e^x \ = \ 8\)

7) Solve for \(x\): \(ln \ x \ = \ 11\)

8) Solve for \(x\): \(ln \ (x \ + \ 5) \ = \ 20\)

9) Solve for \(x\): \(ln \ (3x \ - \ 9) \ = \ 17\)

10) Solve for \(x\): \(ln \ x \ = \ 2 \ ln \ 5 \ + \ 2 \ ln \ 2\)

 

1) Evaluate without using a calculator: \(3 \ ln \ e\)

\(\color{red}{ln \ e \ = \ log_e \ e \ = \ 1 \ ⇒ \ 3 \ ln \ e \ = \ 3}\)

2) Evaluate without using a calculator: \(-3 \ ln \ e^5\)

\(\color{red}{ln \ e \ = \ log_e \ e \ = \ 1 \ ⇒ \ -3 \ ln \ e^5 \ = \ -3 \times 5 \ ln \ e \ = \ -15}\)

3) Evaluate without using a calculator: \(9 \ ln \ \frac{1}{e}\)

\(\color{red}{ln \ e \ = \ log_e \ e \ = \ 1 \ ⇒ \ 9 \ ln \ \frac{1}{e} \ = \ -9 \ ln \ e \ = \ -9}\)

4) Evaluate without using a calculator: \(6 \ ln \ \frac{1}{e^5}\)

\(\color{red}{ln \ e \ = \ log_e \ e \ = \ 1 \ ⇒ \ 6 \ ln \ \frac{1}{e^5} \ = \ -6 \ ln \ e^5 \ = \ -6 \times 5 \ ln \ e \ = \ -30}\)

5) Evaluate without using a calculator: \(e^{ln \ \frac{1}{e}}\)

\(\color{red}{e^{ln \ \frac{1}{e}} \ = \ \frac{1}{e}}\)

6) Solve for \(x\): \(e^x \ = \ 8\)

\(\color{red}{e^x \ = \ 8 \ ⇒ \ ln \ e^x \ = \ ln \ 8 \ ⇒ \ x \ ln \ e \ = \ ln \ 8 \ ⇒ \ x \ = \ \frac{ln \ 8}{ln \ e} \ = \ ln \ 8}\)

7) Solve for \(x\): \(ln \ x \ = \ 11\)

\(\color{red}{ln \ x \ = \ 11 \ ⇒ \ e^{ln \ x} \ = \ e^{11} \ ⇒ \ x \ = \ e^{11}}\)

8) Solve for \(x\): \(ln \ (x \ + \ 5) \ = \ 20\)

\(\color{red}{ln \ (x \ + \ 5) \ = \ 20 \ ⇒ \ e^{ln \ (x \ + \ 5)} \ = \ e^{20} \ ⇒ \ x \ + \ 5 \ = \ e^{20} \ ⇒ \ x \ = \ e^{20} \ - \ 5}\)

9) Solve for \(x\): \(ln \ (3x \ - \ 9) \ = \ 17\)

\(\color{red}{ln \ (3x \ - \ 9) \ = \ 17 \ ⇒ \ e^{ln \ (3x \ - \ 9)} \ = \ e^{17} \ ⇒ \ 3x \ - \ 9 \ = \ e^{17} \ ⇒ \ 3x \ = \ e^{17} \ + \ 9 \ ⇒}\) \(\color{red}{x \ = \ \frac{e^{17} \  +\ 9}{3} \ = \ x \ = \ \frac{1}{3} \ e^{17} \ + \ 3}\)

10) Solve for \(x\): \(ln \ x \ = \ 2 \ ln \ 5 \ + \ 2 \ ln \ 2\)

\(\color{red}{ln \ x \ = \ 2 \ ln \ 5 \ + \ 2 \ ln \ 2 \ ⇒ \ e^{ln \ x} \ = \ e^{2 \ ln \ 5 \ + \ 2 \ ln \ 2} \ ⇒ \ e^{ln \ x} \ = \ e^{ln \ 5^2 \times 2^2} \ ⇒ }\)\(\color{red}{ \ x \ = \ 5^2 \times 2^2 \ = \ 100}\)

Natural Logarithms Practice Quiz