# 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 $\text{\hspace{0.17em}}x?$

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
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
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
find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
x=-b+_Гb2-(4ac) ______________ 2a
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator