# 4.5 Logarithmic properties  (Page 6/10)

 Page 6 / 10

Given a logarithm with the form $\text{\hspace{0.17em}}{\mathrm{log}}_{b}M,$ use the change-of-base formula to rewrite it as a quotient of logs with any positive base $\text{\hspace{0.17em}}n,$ where $\text{\hspace{0.17em}}n\ne 1.$

1. Determine the new base $\text{\hspace{0.17em}}n,$ remembering that the common log, $\text{\hspace{0.17em}}\mathrm{log}\left(x\right),$ has base 10, and the natural log, $\text{\hspace{0.17em}}\mathrm{ln}\left(x\right),$ has base $\text{\hspace{0.17em}}e.$
2. Rewrite the log as a quotient using the change-of-base formula
• The numerator of the quotient will be a logarithm with base $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ and argument $\text{\hspace{0.17em}}M.$
• The denominator of the quotient will be a logarithm with base $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ and argument $\text{\hspace{0.17em}}b.$

## Changing logarithmic expressions to expressions involving only natural logs

Change $\text{\hspace{0.17em}}{\mathrm{log}}_{5}3\text{\hspace{0.17em}}$ to a quotient of natural logarithms.

Because we will be expressing $\text{\hspace{0.17em}}{\mathrm{log}}_{5}3\text{\hspace{0.17em}}$ as a quotient of natural logarithms, the new base, $\text{\hspace{0.17em}}n=e.$

We rewrite the log as a quotient using the change-of-base formula. The numerator of the quotient will be the natural log with argument 3. The denominator of the quotient will be the natural log with argument 5.

$\begin{array}{ll}{\mathrm{log}}_{b}M\hfill & =\frac{\mathrm{ln}M}{\mathrm{ln}b}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{log}}_{5}3\hfill & =\frac{\mathrm{ln}3}{\mathrm{ln}5}\hfill \end{array}$

Change $\text{\hspace{0.17em}}{\mathrm{log}}_{0.5}8\text{\hspace{0.17em}}$ to a quotient of natural logarithms.

$\frac{\mathrm{ln}8}{\mathrm{ln}0.5}$

Can we change common logarithms to natural logarithms?

Yes. Remember that $\text{\hspace{0.17em}}\mathrm{log}9\text{\hspace{0.17em}}$ means $\text{\hspace{0.17em}}{\text{log}}_{\text{10}}\text{9}.$ So, $\text{\hspace{0.17em}}\mathrm{log}9=\frac{\mathrm{ln}9}{\mathrm{ln}10}.$

## Using the change-of-base formula with a calculator

Evaluate $\text{\hspace{0.17em}}{\mathrm{log}}_{2}\left(10\right)\text{\hspace{0.17em}}$ using the change-of-base formula with a calculator.

According to the change-of-base formula, we can rewrite the log base 2 as a logarithm of any other base. Since our calculators can evaluate the natural log, we might choose to use the natural logarithm, which is the log base $\text{\hspace{0.17em}}e.$

Evaluate $\text{\hspace{0.17em}}{\mathrm{log}}_{5}\left(100\right)\text{\hspace{0.17em}}$ using the change-of-base formula.

$\frac{\mathrm{ln}100}{\mathrm{ln}5}\approx \frac{4.6051}{1.6094}=2.861$

Access these online resources for additional instruction and practice with laws of logarithms.

## Key equations

 The Product Rule for Logarithms ${\mathrm{log}}_{b}\left(MN\right)={\mathrm{log}}_{b}\left(M\right)+{\mathrm{log}}_{b}\left(N\right)$ The Quotient Rule for Logarithms ${\mathrm{log}}_{b}\left(\frac{M}{N}\right)={\mathrm{log}}_{b}M-{\mathrm{log}}_{b}N$ The Power Rule for Logarithms ${\mathrm{log}}_{b}\left({M}^{n}\right)=n{\mathrm{log}}_{b}M$ The Change-of-Base Formula

## Key concepts

• We can use the product rule of logarithms to rewrite the log of a product as a sum of logarithms. See [link] .
• We can use the quotient rule of logarithms to rewrite the log of a quotient as a difference of logarithms. See [link] .
• We can use the power rule for logarithms to rewrite the log of a power as the product of the exponent and the log of its base. See [link] , [link] , and [link] .
• We can use the product rule, the quotient rule, and the power rule together to combine or expand a logarithm with a complex input. See [link] , [link] , and [link] .
• The rules of logarithms can also be used to condense sums, differences, and products with the same base as a single logarithm. See [link] , [link] , [link] , and [link] .
• We can convert a logarithm with any base to a quotient of logarithms with any other base using the change-of-base formula. See [link] .
• The change-of-base formula is often used to rewrite a logarithm with a base other than 10 and $\text{\hspace{0.17em}}e\text{\hspace{0.17em}}$ as the quotient of natural or common logs. That way a calculator can be used to evaluate. See [link] .

## Verbal

How does the power rule for logarithms help when solving logarithms with the form $\text{\hspace{0.17em}}{\mathrm{log}}_{b}\left(\sqrt[n]{x}\right)?$

Any root expression can be rewritten as an expression with a rational exponent so that the power rule can be applied, making the logarithm easier to calculate. Thus, $\text{\hspace{0.17em}}{\mathrm{log}}_{b}\left({x}^{\frac{1}{n}}\right)=\frac{1}{n}{\mathrm{log}}_{b}\left(x\right).$

What does the change-of-base formula do? Why is it useful when using a calculator?

## Algebraic

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

${\mathrm{log}}_{b}\left(7x\cdot 2y\right)$

${\mathrm{log}}_{b}\left(2\right)+{\mathrm{log}}_{b}\left(7\right)+{\mathrm{log}}_{b}\left(x\right)+{\mathrm{log}}_{b}\left(y\right)$

$\mathrm{ln}\left(3ab\cdot 5c\right)$

${\mathrm{log}}_{b}\left(\frac{13}{17}\right)$

${\mathrm{log}}_{b}\left(13\right)-{\mathrm{log}}_{b}\left(17\right)$

$\mathrm{ln}\left(\frac{1}{{4}^{k}}\right)$

$-k\mathrm{ln}\left(4\right)$

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

For the following exercises, condense to a single logarithm if possible.

$\mathrm{ln}\left(7\right)+\mathrm{ln}\left(x\right)+\mathrm{ln}\left(y\right)$

$\mathrm{ln}\left(7xy\right)$

${\mathrm{log}}_{3}\left(2\right)+{\mathrm{log}}_{3}\left(a\right)+{\mathrm{log}}_{3}\left(11\right)+{\mathrm{log}}_{3}\left(b\right)$

${\mathrm{log}}_{b}\left(28\right)-{\mathrm{log}}_{b}\left(7\right)$

${\mathrm{log}}_{b}\left(4\right)$

$\mathrm{ln}\left(a\right)-\mathrm{ln}\left(d\right)-\mathrm{ln}\left(c\right)$

$-{\mathrm{log}}_{b}\left(\frac{1}{7}\right)$

${\text{log}}_{b}\left(7\right)$

$\frac{1}{3}\mathrm{ln}\left(8\right)$

For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

$\mathrm{log}\left(\frac{{x}^{15}{y}^{13}}{{z}^{19}}\right)$

$15\mathrm{log}\left(x\right)+13\mathrm{log}\left(y\right)-19\mathrm{log}\left(z\right)$

$\mathrm{ln}\left(\frac{{a}^{-2}}{{b}^{-4}{c}^{5}}\right)$

$\mathrm{log}\left(\sqrt{{x}^{3}{y}^{-4}}\right)$

$\frac{3}{2}\mathrm{log}\left(x\right)-2\mathrm{log}\left(y\right)$

$\mathrm{ln}\left(y\sqrt{\frac{y}{1-y}}\right)$

$\mathrm{log}\left({x}^{2}{y}^{3}\sqrt[3]{{x}^{2}{y}^{5}}\right)$

$\frac{8}{3}\mathrm{log}\left(x\right)+\frac{14}{3}\mathrm{log}\left(y\right)$

For the following exercises, condense each expression to a single logarithm using the properties of logarithms.

$\mathrm{log}\left(2{x}^{4}\right)+\mathrm{log}\left(3{x}^{5}\right)$

$\mathrm{ln}\left(6{x}^{9}\right)-\mathrm{ln}\left(3{x}^{2}\right)$

$\mathrm{ln}\left(2{x}^{7}\right)$

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

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

$\mathrm{log}\left(\frac{x{z}^{3}}{\sqrt{y}}\right)$

$4{\mathrm{log}}_{7}\left(c\right)+\frac{{\mathrm{log}}_{7}\left(a\right)}{3}+\frac{{\mathrm{log}}_{7}\left(b\right)}{3}$

For the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base.

${\mathrm{log}}_{7}\left(15\right)\text{\hspace{0.17em}}$ to base $\text{\hspace{0.17em}}e$

${\mathrm{log}}_{7}\left(15\right)=\frac{\mathrm{ln}\left(15\right)}{\mathrm{ln}\left(7\right)}$

${\mathrm{log}}_{14}\left(55.875\right)\text{\hspace{0.17em}}$ to base $\text{\hspace{0.17em}}10$

For the following exercises, suppose $\text{\hspace{0.17em}}{\mathrm{log}}_{5}\left(6\right)=a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{\mathrm{log}}_{5}\left(11\right)=b.\text{\hspace{0.17em}}$ Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b.\text{\hspace{0.17em}}$ Show the steps for solving.

${\mathrm{log}}_{11}\left(5\right)$

${\mathrm{log}}_{11}\left(5\right)=\frac{{\mathrm{log}}_{5}\left(5\right)}{{\mathrm{log}}_{5}\left(11\right)}=\frac{1}{b}$

${\mathrm{log}}_{6}\left(55\right)$

${\mathrm{log}}_{11}\left(\frac{6}{11}\right)$

${\mathrm{log}}_{11}\left(\frac{6}{11}\right)=\frac{{\mathrm{log}}_{5}\left(\frac{6}{11}\right)}{{\mathrm{log}}_{5}\left(11\right)}=\frac{{\mathrm{log}}_{5}\left(6\right)-{\mathrm{log}}_{5}\left(11\right)}{{\mathrm{log}}_{5}\left(11\right)}=\frac{a-b}{b}=\frac{a}{b}-1$

## Numeric

For the following exercises, use properties of logarithms to evaluate without using a calculator.

${\mathrm{log}}_{3}\left(\frac{1}{9}\right)-3{\mathrm{log}}_{3}\left(3\right)$

$6{\mathrm{log}}_{8}\left(2\right)+\frac{{\mathrm{log}}_{8}\left(64\right)}{3{\mathrm{log}}_{8}\left(4\right)}$

$3$

$2{\mathrm{log}}_{9}\left(3\right)-4{\mathrm{log}}_{9}\left(3\right)+{\mathrm{log}}_{9}\left(\frac{1}{729}\right)$

For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places.

${\mathrm{log}}_{3}\left(22\right)$

$2.81359$

${\mathrm{log}}_{8}\left(65\right)$

${\mathrm{log}}_{6}\left(5.38\right)$

$0.93913$

${\mathrm{log}}_{4}\left(\frac{15}{2}\right)$

${\mathrm{log}}_{\frac{1}{2}}\left(4.7\right)$

$-2.23266$

## Extensions

Use the product rule for logarithms to find all $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ values such that $\text{\hspace{0.17em}}{\mathrm{log}}_{12}\left(2x+6\right)+{\mathrm{log}}_{12}\left(x+2\right)=2.\text{\hspace{0.17em}}$ Show the steps for solving.

Use the quotient rule for logarithms to find all $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ values such that $\text{\hspace{0.17em}}{\mathrm{log}}_{6}\left(x+2\right)-{\mathrm{log}}_{6}\left(x-3\right)=1.\text{\hspace{0.17em}}$ Show the steps for solving.

$x=4;\text{\hspace{0.17em}}$ By the quotient rule: ${\mathrm{log}}_{6}\left(x+2\right)-{\mathrm{log}}_{6}\left(x-3\right)={\mathrm{log}}_{6}\left(\frac{x+2}{x-3}\right)=1.$

Rewriting as an exponential equation and solving for $\text{\hspace{0.17em}}x:$

$\begin{array}{ll}{6}^{1}\hfill & =\frac{x+2}{x-3}\hfill \\ \text{\hspace{0.17em}}0\hfill & =\frac{x+2}{x-3}-6\hfill \\ \text{\hspace{0.17em}}0\hfill & =\frac{x+2}{x-3}-\frac{6\left(x-3\right)}{\left(x-3\right)}\hfill \\ \text{\hspace{0.17em}}0\hfill & =\frac{x+2-6x+18}{x-3}\hfill \\ \text{\hspace{0.17em}}0\hfill & =\frac{x-4}{x-3}\hfill \\ \text{​}\text{\hspace{0.17em}}x\hfill & =4\hfill \end{array}$

Checking, we find that $\text{\hspace{0.17em}}{\mathrm{log}}_{6}\left(4+2\right)-{\mathrm{log}}_{6}\left(4-3\right)={\mathrm{log}}_{6}\left(6\right)-{\mathrm{log}}_{6}\left(1\right)\text{\hspace{0.17em}}$ is defined, so $\text{\hspace{0.17em}}x=4.$

Can the power property of logarithms be derived from the power property of exponents using the equation $\text{\hspace{0.17em}}{b}^{x}=m?\text{\hspace{0.17em}}$ If not, explain why. If so, show the derivation.

Prove that $\text{\hspace{0.17em}}{\mathrm{log}}_{b}\left(n\right)=\frac{1}{{\mathrm{log}}_{n}\left(b\right)}\text{\hspace{0.17em}}$ for any positive integers $\text{\hspace{0.17em}}b>1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}n>1.$

Let $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ be positive integers greater than $\text{\hspace{0.17em}}1.\text{\hspace{0.17em}}$ Then, by the change-of-base formula, $\text{\hspace{0.17em}}{\mathrm{log}}_{b}\left(n\right)=\frac{{\mathrm{log}}_{n}\left(n\right)}{{\mathrm{log}}_{n}\left(b\right)}=\frac{1}{{\mathrm{log}}_{n}\left(b\right)}.$

Does $\text{\hspace{0.17em}}{\mathrm{log}}_{81}\left(2401\right)={\mathrm{log}}_{3}\left(7\right)?\text{\hspace{0.17em}}$ Verify the claim algebraically.

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
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
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
I had just woken up when i got this message
Joe
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
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
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert
can I see the picture