What is Natural Logarithm

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 $$x \ > \ 0$$,

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

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 $$x$$.

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 $$x$$: $$ln \ x \ = \ 2 \ ln \ 5 \ + \ 2 \ ln \ 2$$

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

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