# 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] .

give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
can you not take the square root of a negative number
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena