# 4.5 Logarithmic properties  (Page 3/10)

 Page 3 / 10

## 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:

#### Questions & Answers

The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
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Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
divide simplify each answer 3/2÷5/4
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
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
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
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