# 4.3 Logarithmic functions  (Page 3/9)

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Write the following exponential equations in logarithmic form.

1. ${3}^{2}=9$
2. ${5}^{3}=125$
3. ${2}^{-1}=\frac{1}{2}$
1. ${3}^{2}=9\text{\hspace{0.17em}}$ is equivalent to $\text{\hspace{0.17em}}{\mathrm{log}}_{3}\left(9\right)=2$
2. ${5}^{3}=125\text{\hspace{0.17em}}$ is equivalent to $\text{\hspace{0.17em}}{\mathrm{log}}_{5}\left(125\right)=3$
3. ${2}^{-1}=\frac{1}{2}\text{\hspace{0.17em}}$ is equivalent to $\text{\hspace{0.17em}}{\text{log}}_{2}\left(\frac{1}{2}\right)=-1$

## Evaluating logarithms

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider $\text{\hspace{0.17em}}{\mathrm{log}}_{2}8.\text{\hspace{0.17em}}$ We ask, “To what exponent must $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}$ be raised in order to get 8?” Because we already know $\text{\hspace{0.17em}}{2}^{3}=8,$ it follows that $\text{\hspace{0.17em}}{\mathrm{log}}_{2}8=3.$

Now consider solving $\text{\hspace{0.17em}}{\mathrm{log}}_{7}49\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{\mathrm{log}}_{3}27\text{\hspace{0.17em}}$ mentally.

• We ask, “To what exponent must 7 be raised in order to get 49?” We know $\text{\hspace{0.17em}}{7}^{2}=49.\text{\hspace{0.17em}}$ Therefore, $\text{\hspace{0.17em}}{\mathrm{log}}_{7}49=2$
• We ask, “To what exponent must 3 be raised in order to get 27?” We know $\text{\hspace{0.17em}}{3}^{3}=27.\text{\hspace{0.17em}}$ Therefore, $\text{\hspace{0.17em}}{\mathrm{log}}_{3}27=3$

Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate $\text{\hspace{0.17em}}{\mathrm{log}}_{\frac{2}{3}}\frac{4}{9}\text{\hspace{0.17em}}$ mentally.

• We ask, “To what exponent must $\text{\hspace{0.17em}}\frac{2}{3}\text{\hspace{0.17em}}$ be raised in order to get $\text{\hspace{0.17em}}\frac{4}{9}?\text{\hspace{0.17em}}$ ” We know $\text{\hspace{0.17em}}{2}^{2}=4\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{3}^{2}=9,$ so $\text{\hspace{0.17em}}{\left(\frac{2}{3}\right)}^{2}=\frac{4}{9}.\text{\hspace{0.17em}}$ Therefore, $\text{\hspace{0.17em}}{\mathrm{log}}_{\frac{2}{3}}\left(\frac{4}{9}\right)=2.$

Given a logarithm of the form $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right),$ evaluate it mentally.

1. Rewrite the argument $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ as a power of $\text{\hspace{0.17em}}b:\text{\hspace{0.17em}}$ ${b}^{y}=x.\text{\hspace{0.17em}}$
2. Use previous knowledge of powers of $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ identify $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ by asking, “To what exponent should $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ be raised in order to get $\text{\hspace{0.17em}}x?$

## Solving logarithms mentally

Solve $\text{\hspace{0.17em}}y={\mathrm{log}}_{4}\left(64\right)\text{\hspace{0.17em}}$ without using a calculator.

First we rewrite the logarithm in exponential form: $\text{\hspace{0.17em}}{4}^{y}=64.\text{\hspace{0.17em}}$ Next, we ask, “To what exponent must 4 be raised in order to get 64?”

We know

${4}^{3}=64$

Therefore,

$\mathrm{log}{}_{4}\left(64\right)=3$

Solve $\text{\hspace{0.17em}}y={\mathrm{log}}_{121}\left(11\right)\text{\hspace{0.17em}}$ without using a calculator.

${\mathrm{log}}_{121}\left(11\right)=\frac{1}{2}\text{\hspace{0.17em}}$ (recalling that $\text{\hspace{0.17em}}\sqrt{121}={\left(121\right)}^{\frac{1}{2}}=11$ )

## Evaluating the logarithm of a reciprocal

Evaluate $\text{\hspace{0.17em}}y={\mathrm{log}}_{3}\left(\frac{1}{27}\right)\text{\hspace{0.17em}}$ without using a calculator.

First we rewrite the logarithm in exponential form: $\text{\hspace{0.17em}}{3}^{y}=\frac{1}{27}.\text{\hspace{0.17em}}$ Next, we ask, “To what exponent must 3 be raised in order to get $\text{\hspace{0.17em}}\frac{1}{27}?$

We know $\text{\hspace{0.17em}}{3}^{3}=27,$ but what must we do to get the reciprocal, $\text{\hspace{0.17em}}\frac{1}{27}?\text{\hspace{0.17em}}$ Recall from working with exponents that $\text{\hspace{0.17em}}{b}^{-a}=\frac{1}{{b}^{a}}.\text{\hspace{0.17em}}$ We use this information to write

$\begin{array}{l}{3}^{-3}=\frac{1}{{3}^{3}}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{1}{27}\hfill \end{array}$

Therefore, $\text{\hspace{0.17em}}{\mathrm{log}}_{3}\left(\frac{1}{27}\right)=-3.$

Evaluate $\text{\hspace{0.17em}}y={\mathrm{log}}_{2}\left(\frac{1}{32}\right)\text{\hspace{0.17em}}$ without using a calculator.

${\mathrm{log}}_{2}\left(\frac{1}{32}\right)=-5$

## Using common logarithms

Sometimes we may see a logarithm written without a base. In this case, we assume that the base is 10. In other words, the expression $\text{\hspace{0.17em}}\mathrm{log}\left(x\right)\text{\hspace{0.17em}}$ means $\text{\hspace{0.17em}}{\mathrm{log}}_{10}\left(x\right).\text{\hspace{0.17em}}$ We call a base-10 logarithm a common logarithm . Common logarithms are used to measure the Richter Scale mentioned at the beginning of the section. Scales for measuring the brightness of stars and the pH of acids and bases also use common logarithms.

## Definition of the common logarithm

A common logarithm    is a logarithm with base $\text{\hspace{0.17em}}10.\text{\hspace{0.17em}}$ We write $\text{\hspace{0.17em}}{\mathrm{log}}_{10}\left(x\right)\text{\hspace{0.17em}}$ simply as $\text{\hspace{0.17em}}\mathrm{log}\left(x\right).\text{\hspace{0.17em}}$ The common logarithm of a positive number $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ satisfies the following definition.

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

We read $\text{\hspace{0.17em}}\mathrm{log}\left(x\right)\text{\hspace{0.17em}}$ as, “the logarithm with base $\text{\hspace{0.17em}}10\text{\hspace{0.17em}}$ of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ ” or “log base 10 of $\text{\hspace{0.17em}}x.$

The logarithm $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ is the exponent to which $\text{\hspace{0.17em}}10\text{\hspace{0.17em}}$ must be raised to get $\text{\hspace{0.17em}}x.$

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

1. Rewrite the argument $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ as a power of $\text{\hspace{0.17em}}10:\text{\hspace{0.17em}}$ ${10}^{y}=x.$
2. Use previous knowledge of powers of $\text{\hspace{0.17em}}10\text{\hspace{0.17em}}$ to identify $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ by asking, “To what exponent must $\text{\hspace{0.17em}}10\text{\hspace{0.17em}}$ be raised in order to get $\text{\hspace{0.17em}}x?$

what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
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