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Simplify Natural Logarithms Solutions

•    e    is defined as: $e=\underset{n\to \infty }{lim} {\left(1+\frac{1}{n}\right)}^n$

• This irrational number $\underline{ \mathit{e}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{718281}\mathbf{\dots }\mathbf{ }\mathbf{ }\mathbf{ }}$ has an approximate value of 2.7182, which is used when working with natural logarithms.
• 
The natural log is written either of these ways:
  • 
$\underline{ {\mathbf{log}}_{\mathit{e}}\mathbf{ }\mathit{x}\mathbf{ }\mathbf{ }\mathbf{ }}$
  • 
$\underline{ \mathbf{ln}\mathbf{ }\mathit{x}\mathbf{ }\mathbf{ }\mathbf{ }}$

• With ln:

  • The letter “l” stands for    logarithm   .

  • The letter “n” stands for    natural   .

Properties of Logs

For all rules of logs, the variables a, b, and c are positive real numbers, n and x are real numbers, and $a\ne 1.$

Properties of Logs	Logarithm Rule(s)	Natural Logarithm
Foundational Properties	If $a^0=1,$ then ${\log }_{a}1=0$	$\ln 1=0$
	If $a^n=a^n,$ then ${\log }_a\left(a^n\right)=n$	$\ln \left(e^x\right)=x \mathrm{and} e^{\ln x}=x$
	If $a^1=a,$ then ${\log }_a\left(a\right)=1$	$\ln e=1$
Quotient Rule	${\log }_a\frac{b}{c}={\log }_{a}b–{\log }_{a}c$	$\ln \frac{b}{c}=\ln b–\ln c$
Product Rule	${\log }_{a}bc={\log }_{a}b+{\log }_{a}c$	$\ln bc=\ln b+\ln c$
Power Rule	${\log }_a b^n=n·{\log }_a b$	$\ln b^n=n·\ln b$

Example 1

Evaluate the expression.

Note

Problems A–C

In previous lessons, you have evaluated similar expressions with common logs as you have solved problems. For example, $\log 100=2$ since $\log {10}^2=2$.