# 4.5 Logarithmic properties  (Page 3/10)

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## Using the quotient rule for logarithms

Expand $\text{\hspace{0.17em}}{\mathrm{log}}_{2}\left(\frac{15x\left(x-1\right)}{\left(3x+4\right)\left(2-x\right)}\right).$

First we note that the quotient is factored and in lowest terms, so we apply the quotient rule.

${\mathrm{log}}_{2}\left(\frac{15x\left(x-1\right)}{\left(3x+4\right)\left(2-x\right)}\right)={\mathrm{log}}_{2}\left(15x\left(x-1\right)\right)-{\mathrm{log}}_{2}\left(\left(3x+4\right)\left(2-x\right)\right)$

Notice that the resulting terms are logarithms of products. To expand completely, we apply the product rule, noting that the prime factors of the factor 15 are 3 and 5.

Expand $\text{\hspace{0.17em}}{\mathrm{log}}_{3}\left(\frac{7{x}^{2}+21x}{7x\left(x-1\right)\left(x-2\right)}\right).$

${\mathrm{log}}_{3}\left(x+3\right)-{\mathrm{log}}_{3}\left(x-1\right)-{\mathrm{log}}_{3}\left(x-2\right)$

## Using the power rule for logarithms

We’ve explored the product rule and the quotient rule, but how can we take the logarithm of a power, such as $\text{\hspace{0.17em}}{x}^{2}?\text{\hspace{0.17em}}$ One method is as follows:

$\begin{array}{ll}{\mathrm{log}}_{b}\left({x}^{2}\right)\hfill & ={\mathrm{log}}_{b}\left(x\cdot x\right)\hfill \\ \hfill & ={\mathrm{log}}_{b}x+{\mathrm{log}}_{b}x\hfill \\ \hfill & =2{\mathrm{log}}_{b}x\hfill \end{array}$

Notice that we used the product rule for logarithms    to find a solution for the example above. By doing so, we have derived the power rule for logarithms , which says that the log of a power is equal to the exponent times the log of the base. Keep in mind that, although the input to a logarithm may not be written as a power, we may be able to change it to a power. For example,

$\begin{array}{lll}100={10}^{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\hfill & \sqrt{3}={3}^{\frac{1}{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\hfill & \frac{1}{e}={e}^{-1}\hfill \end{array}$

## The power rule for logarithms

The power rule for logarithms    can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.

${\mathrm{log}}_{b}\left({M}^{n}\right)=n{\mathrm{log}}_{b}M$

Given the logarithm of a power, use the power rule of logarithms to write an equivalent product of a factor and a logarithm.

1. Express the argument as a power, if needed.
2. Write the equivalent expression by multiplying the exponent times the logarithm of the base.

## Expanding a logarithm with powers

Expand $\text{\hspace{0.17em}}{\mathrm{log}}_{2}{x}^{5}.$

The argument is already written as a power, so we identify the exponent, 5, and the base, $\text{\hspace{0.17em}}x,$ and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.

${\mathrm{log}}_{2}\left({x}^{5}\right)=5{\mathrm{log}}_{2}x$

Expand $\text{\hspace{0.17em}}\mathrm{ln}{x}^{2}.\text{\hspace{0.17em}}$

$2\mathrm{ln}x$

## Rewriting an expression as a power before using the power rule

Expand $\text{\hspace{0.17em}}{\mathrm{log}}_{3}\left(25\right)\text{\hspace{0.17em}}$ using the power rule for logs.

Expressing the argument as a power, we get $\text{\hspace{0.17em}}{\mathrm{log}}_{3}\left(25\right)={\mathrm{log}}_{3}\left({5}^{2}\right).$

Next we identify the exponent, 2, and the base, 5, and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.

${\mathrm{log}}_{3}\left({5}^{2}\right)=2{\mathrm{log}}_{3}\left(5\right)$

Expand $\text{\hspace{0.17em}}\mathrm{ln}\left(\frac{1}{{x}^{2}}\right).$

$-2\mathrm{ln}\left(x\right)$

## Using the power rule in reverse

Rewrite $\text{\hspace{0.17em}}4\mathrm{ln}\left(x\right)\text{\hspace{0.17em}}$ using the power rule for logs to a single logarithm with a leading coefficient of 1.

Because the logarithm of a power is the product of the exponent times the logarithm of the base, it follows that the product of a number and a logarithm can be written as a power. For the expression $\text{\hspace{0.17em}}4\mathrm{ln}\left(x\right),$ we identify the factor, 4, as the exponent and the argument, $\text{\hspace{0.17em}}x,$ as the base, and rewrite the product as a logarithm of a power: $\text{\hspace{0.17em}}4\mathrm{ln}\left(x\right)=\mathrm{ln}\left({x}^{4}\right).\text{\hspace{0.17em}}$

Rewrite $\text{\hspace{0.17em}}2{\mathrm{log}}_{3}4\text{\hspace{0.17em}}$ using the power rule for logs to a single logarithm with a leading coefficient of 1.

${\mathrm{log}}_{3}16$

## Expanding logarithmic expressions

Taken together, the product rule, quotient rule, and power rule are often called “laws of logs.” Sometimes we apply more than one rule in order to simplify an expression. For example:

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
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