# 4.3 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] .

#### Questions & Answers

how to understand calculus?
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
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what type of identity
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Confunction Identity
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meena
hello guys
meena
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It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
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