# 6.3 Logarithmic functions  (Page 4/9)

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## Finding the value of a common logarithm mentally

Evaluate $\text{\hspace{0.17em}}y=\mathrm{log}\left(1000\right)\text{\hspace{0.17em}}$ without using a calculator.

First we rewrite the logarithm in exponential form: $\text{\hspace{0.17em}}{10}^{y}=1000.\text{\hspace{0.17em}}$ Next, we ask, “To what exponent must $\text{\hspace{0.17em}}10\text{\hspace{0.17em}}$ be raised in order to get 1000?” We know

${10}^{3}=1000$

Therefore, $\text{\hspace{0.17em}}\mathrm{log}\left(1000\right)=3.$

Evaluate $\text{\hspace{0.17em}}y=\mathrm{log}\left(1,000,000\right).$

$\mathrm{log}\left(1,000,000\right)=6$

Given a common logarithm with the form $\text{\hspace{0.17em}}y=\mathrm{log}\left(x\right),$ evaluate it using a calculator.

1. Press [LOG] .
2. Enter the value given for $\text{\hspace{0.17em}}x,$ followed by [ ) ] .
3. Press [ENTER] .

## Finding the value of a common logarithm using a calculator

Evaluate $\text{\hspace{0.17em}}y=\mathrm{log}\left(321\right)\text{\hspace{0.17em}}$ to four decimal places using a calculator.

• Press [LOG] .
• Enter 321 , followed by [ ) ] .
• Press [ENTER] .

Rounding to four decimal places, $\text{\hspace{0.17em}}\mathrm{log}\left(321\right)\approx 2.5065.$

Evaluate $\text{\hspace{0.17em}}y=\mathrm{log}\left(123\right)\text{\hspace{0.17em}}$ to four decimal places using a calculator.

$\mathrm{log}\left(123\right)\approx 2.0899$

## Rewriting and solving a real-world exponential model

The amount of energy released from one earthquake was 500 times greater than the amount of energy released from another. The equation $\text{\hspace{0.17em}}{10}^{x}=500\text{\hspace{0.17em}}$ represents this situation, where $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

We begin by rewriting the exponential equation in logarithmic form.

Next we evaluate the logarithm using a calculator:

• Press [LOG] .
• Enter $\text{\hspace{0.17em}}500,$ followed by [ ) ] .
• Press [ENTER] .
• To the nearest thousandth, $\text{\hspace{0.17em}}\mathrm{log}\left(500\right)\approx 2.699.$

The difference in magnitudes was about $\text{\hspace{0.17em}}2.699.$

The amount of energy released from one earthquake was $\text{\hspace{0.17em}}\text{8,500}\text{\hspace{0.17em}}$ times greater than the amount of energy released from another. The equation $\text{\hspace{0.17em}}{10}^{x}=8500\text{\hspace{0.17em}}$ represents this situation, where $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

The difference in magnitudes was about $\text{\hspace{0.17em}}3.929.$

## Using natural logarithms

The most frequently used base for logarithms is $\text{\hspace{0.17em}}e.\text{\hspace{0.17em}}$ Base $\text{\hspace{0.17em}}e\text{\hspace{0.17em}}$ logarithms are important in calculus and some scientific applications; they are called natural logarithms . The base $\text{\hspace{0.17em}}e\text{\hspace{0.17em}}$ logarithm, $\text{\hspace{0.17em}}{\mathrm{log}}_{e}\left(x\right),$ has its own notation, $\text{\hspace{0.17em}}\mathrm{ln}\left(x\right).$

Most values of $\text{\hspace{0.17em}}\mathrm{ln}\left(x\right)\text{\hspace{0.17em}}$ can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, $\text{\hspace{0.17em}}\mathrm{ln}1=0.\text{\hspace{0.17em}}$ For other natural logarithms, we can use the $\text{\hspace{0.17em}}\mathrm{ln}\text{\hspace{0.17em}}$ key that can be found on most scientific calculators. We can also find the natural logarithm of any power of $\text{\hspace{0.17em}}e\text{\hspace{0.17em}}$ using the inverse property of logarithms.

## Definition of the natural logarithm

A natural logarithm    is a logarithm with base $\text{\hspace{0.17em}}e.$ We write ${\mathrm{log}}_{e}\left(x\right)$ simply as $\mathrm{ln}\left(x\right).$ The natural logarithm of a positive number $x$ satisfies the following definition.

For $\text{\hspace{0.17em}}x>0,$

We read $\text{\hspace{0.17em}}\mathrm{ln}\left(x\right)\text{\hspace{0.17em}}$ as, “the logarithm with base $\text{\hspace{0.17em}}e\text{\hspace{0.17em}}$ of $\text{\hspace{0.17em}}x$ ” or “the natural logarithm of $\text{\hspace{0.17em}}x.$

The logarithm $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ is the exponent to which $\text{\hspace{0.17em}}e\text{\hspace{0.17em}}$ must be raised to get $\text{\hspace{0.17em}}x.$

Since the functions $\text{\hspace{0.17em}}y=e{}^{x}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y=\mathrm{ln}\left(x\right)\text{\hspace{0.17em}}$ are inverse functions, $\text{\hspace{0.17em}}\mathrm{ln}\left({e}^{x}\right)=x\text{\hspace{0.17em}}$ for all $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}e{}^{\mathrm{ln}\left(x\right)}=x\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}x>0.$

Given a natural logarithm with the form $\text{\hspace{0.17em}}y=\mathrm{ln}\left(x\right),$ evaluate it using a calculator.

1. Press [LN] .
2. Enter the value given for $\text{\hspace{0.17em}}x,$ followed by [ ) ] .
3. Press [ENTER] .

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