Call Now to Set Up Tutoring:

(888) 888-0446

Precalculus : Properties of Logarithms

Study concepts, example questions & explanations for Precalculus

Create An Account

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1169 : Pre Calculus

Solve the following logarithmic equation:

2log(x)=log(2)+log(x+3)

Possible Answers:

x=5

$x=2\pm \sqrt{28}$

$x=1\pm \sqrt{7}$

x=-3

$x=\sqrt{7}$

Correct answer:

$x=1\pm \sqrt{7}$

Explanation:

We must first use some properties of logs to rewrite the equation. First, using the power rule, which says

$b\cdot log(x)=log(b^x)$

we can rewrie the left side of the equation, as below:

log(x^2)=log(2)+log(x+3)

Now, we want to use the product property of logarithms to condense the right side into just one log, as below:

log(x^2)=log(2(x+3))

Because the logs are both base 10, we can simply set the insides equal, like this:

x^2=2(x+3)

Now we have a polynomial to solve.

x^2=2x+6

x^2-2x-6=0

Using the quadratic formula to solve for x

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Report an Error

Example Question #71 : Exponential And Logarithmic Functions

Solve this logarithm for :

$25e^{0.02x}=100$

Possible Answers:

x=75.4

x=86

x=69.3

None of these.

x=89.4

Correct answer:

x=69.3

Explanation:

$25e^{0.02x}=100$

Divide both sides by 25 to isolate the exponential function:

$e^{0.02x}=4$

Take the natural log of both sides:

$\ln (e^{0.02x})=\ln(4)$

Solve for x:

$0.02x=\ln 4$

$x=\frac{ln4}{0.02}=\mathbf{69.3}$

Report an Error

Example Question #71 : Exponential And Logarithmic Functions

Solve the following logarithmic equation.

$log_{2}\,x +log_{2}\,(4-x)=2$

Possible Answers:

x=0

x=2

x=1

x=-4

Correct answer:

x=2

Explanation:

In order to solve the logarithmic equation, we use the following property

$log\,a+log\,b=log\,ab$

As such

$log_{2}\,x +log_{2}\,(4-x)=log_{2}\,x(4-x)=log_2\,4x-x^2=2$

And converting from logarithmic form to exponential form

$log_a\,c=b \Rightarrow a^b=c$

we get

2^2=4x-x^2

Solving for x

x^2-4x+4=0

And because the square of a difference is given as this equation through factoring

(a-b)^2=a^2-2ab+b^2

we have

(x-2)^2=0

which implies

x=2

Report an Error

Example Question #11 : Solve Logarithmic Equations

Express the log in its expanded form:

$\log\bigg(\frac{x^3y}{z}\bigg)$

Possible Answers:

None of the other answers

$\log^{3}(x)+\log(y)-\log(z)$

$3\log(x)+\log(y)- \log(z)$

$3\log(x)+\log(y)+\log(z)$

$\log(x)+\log(y)-\log(z)$

Correct answer:

$3\log(x)+\log(y)- \log(z)$

Explanation:

You need to know the Laws of Logarithms in order to solve this problem. The ones specifically used in this problem are the following:

$\log(xy) = \log (x)+\log(y)$

$\log\bigg(\frac{x}{y}\bigg) = \log(x)-\log(y)$

$\log(x^y) = y\cdot \log(x)$

Let's take this one variable at a time starting with expanding z:

$\log\bigg(\frac{x^3y}{z}\bigg) = \log(x^3y)-\log(z)$

Now y:

$\log(x^3y)-\log(z) = \log(x^3)+\log(y)-\log(z)$

And finally expand x:

$\log(x^3)+\log(y)-\log(z) = 3\log(x)+\log(y)-\log(z)$

Report an Error

Example Question #71 : Exponential And Logarithmic Functions

What is ln(2x)-ln(2) equivalent to?

Possible Answers:

$\ln(2)$

$\ln(x)$

$\frac{\ln(2x)}{2}$

$\ln (2x-2)$

Correct answer:

$\ln(x)$

Explanation:

Using the properties of logarithms,

$\log(a)-\log(b)=\log \bigg( \frac{a}{b}\bigg)$

the expression can be rewritten as

$\ln(2x)-\ln(2)=\ln\bigg(\frac{2x}{2}\bigg)$

which simplifies to $\ln(x)$ .

Report an Error

Example Question #81 : Exponential And Logarithmic Functions

Find the value of the sum of logarithms by condensing the expression.

$\ln(2)+\ln\bigg(\frac{1}{2}\bigg)$

Possible Answers:

Undefined

Correct answer:

Explanation:

By the property of the sum of logarithms,

$\ln(2)+\ln\bigg(\frac{1}{2}\bigg)=\ln\bigg(2\cdot\frac{1}{2}\bigg)=\ln(1)=0$ .

Report an Error

Example Question #81 : Exponential And Logarithmic Functions

Condense the following logarithmic equation:

$3\log (y)+\log (4)+\log (x)-2\log (z)$

Possible Answers:

$\log \left(\frac{xy^3}{2z}\right)$

$\log \left(\frac{4xy^3}{z^2}\right)$

$\log (4xy^3-z^2)$

$\log \left(\frac{2z^2}{xy^3}\right)$

$\log \left(\frac{z^2}{4xy^3}\right)$

Correct answer:

$\log \left(\frac{4xy^3}{z^2}\right)$

Explanation:

We start condensing our expression using the following property, which allows us to express the coefficients of two of our terms as exponents:

$b\log (a)=\log (a^b)$

$3\log (y)+\log (4)+\log (x)-2\log (z)$

$\log(y^3)+\log(4)+\log(x)-\log(z^2)$

Our next step is to use the following property to combine our first three terms:

$\log (a)+\log(b)+\log(c)=\log(abc)$

$\log(y^3)+\log(4)+\log(x)-\log(z^2)$

$\log(4xy^3)-\log(z^2)$

Finally, we can use the following property regarding subtraction of logarithms to obtain the condensed expression:

$\log (a)-\log (b)=\log (\frac{a}{b})$

$\log(4xy^3)-\log(z^2)$

$\log \left(\frac{4xy^3}{z^2}\right)$

Report an Error

Example Question #14 : Solve Logarithmic Equations

What is another way of writing

$\frac{\ln(x^b)}{\ln(y^a)}$ ?

Possible Answers:

$\ln [(x/y)^{\frac{b}{a}}]$

$\frac{\ln(x^{1/a})}{\ln(y^{1/b})}$

$\ln [(x/y)^{b-a}]$

$\frac{\ln(x^{b-a})}{\ln y}$

Correct answer:

$\frac{\ln(x^{1/a})}{\ln(y^{1/b})}$

Explanation:

The correct answer is

$\frac{\ln(x^{1/a})}{\ln(y^{1/b})}$

Properties of logarithms allow us to rewrite $\ln y^a$ and $\ln x^b$ as $\ln y^a=a\ln y$ and $b\ln x$ , respectively. So we have

$\frac{\ln(x^b)}{\ln(y^a)}=\frac{b}{a}\frac{\ln(x)}{\ln(y)}=\frac{1/a}{1/b}\frac{\ln(x)}{\ln(y)}$

Again, we use the logarithm property

$\frac{1}{a}\ln x=\ln x^{1/a}$

to get

$\frac{\ln(x^b)}{\ln(y^a)}=\frac{1/a}{1/b}\frac{\ln(x)}{\ln(y)}=\frac{\ln(x^{1/a})}{\ln(y^{1/b})}$

Report an Error

Example Question #14 : Solve Logarithmic Equations

Write the expression in the most condensed form: $4\ln(x)-\frac{1}{2}\ln(x+4)$

Possible Answers:

$ln\frac{4x}{\sqrt{x+4}}$

None of the other answers.

$ln\frac{4x}{x+4}$

$ln\frac{x}{(x+4)^{\frac{1}{2}}}$

$ln\frac{x^4}{\sqrt{x+4}}$

Correct answer:

$ln\frac{x^4}{\sqrt{x+4}}$

Explanation:

$4\ln(x)-\frac{1}{2}\ln(x+4)$

Use the Power property of Logarithms:

$\ln(x)^4-\ln(x+4)^{\frac{1}{2}}$

Rewrite the fractional exponent:

$\ln(x)^4-\ln\sqrt{(x+4)}$

Condense into a fraction using the Quotient property of Logarithms:

$\mathbf{\ln\frac{x^4}{\sqrt{x+4}}}$

Report an Error

Example Question #18 : Solve Logarithmic Equations

Simplify: $\log_2 (8)-\log_2(15)+\log_2(10)$

Possible Answers:

$\log(3)$

$\log_2(\frac{16}{3})$

$\log_2(3)$

$\log(\frac{16}{3})$

$\log_6(3)$

Correct answer:

$\log_2(\frac{16}{3})$

Explanation:

When logs of the same bases are subtracted, the contents of both logs will be divided with each other. When logs of the same bases are added, then the contents inside the log will be multiplied together.

$\log_2 (8)-\log_2(15)= \log_2(8\div 15)=\log_2(\frac{8}{15})$

$\log_2(\frac{8}{15}) + \log_2(10)=\log_2(\frac{8}{15}\times 10) = \log_2(\frac{16}{3})$

Report an Error

View Pre-Calculus Tutors

Tracy
Certified Tutor

Clarion University of Pennsylvania, Bachelor of Science, Physics.

View Pre-Calculus Tutors

Yaakoub
Certified Tutor

CUNY Queens College, Bachelor of Science, Finance. Vaughn College of Aeronautics and Technology, Bachelor of Science, Electri...

View Pre-Calculus Tutors

Emmanuel
Certified Tutor

Baylor University, Bachelor of Science, Biology, General.

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Statistics Tutors in San Francisco-Bay Area, Spanish Tutors in Los Angeles, GRE Tutors in Philadelphia, English Tutors in Denver, Calculus Tutors in Denver, Reading Tutors in Boston, GMAT Tutors in Seattle, Physics Tutors in Chicago, Calculus Tutors in San Francisco-Bay Area, Spanish Tutors in Seattle

Popular Courses & Classes

Spanish Courses & Classes in Philadelphia, Spanish Courses & Classes in Los Angeles, MCAT Courses & Classes in Dallas Fort Worth, SSAT Courses & Classes in Denver, LSAT Courses & Classes in Phoenix, MCAT Courses & Classes in Houston, Spanish Courses & Classes in Houston, GRE Courses & Classes in Dallas Fort Worth, SSAT Courses & Classes in Los Angeles, ACT Courses & Classes in San Francisco-Bay Area

Popular Test Prep

SSAT Test Prep in Denver, SAT Test Prep in Chicago, SAT Test Prep in New York City, SAT Test Prep in Denver, MCAT Test Prep in San Francisco-Bay Area, ISEE Test Prep in Phoenix, GRE Test Prep in Houston, SSAT Test Prep in New York City, GMAT Test Prep in Phoenix, ACT Test Prep in Los Angeles

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Precalculus : Properties of Logarithms

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1169 : Pre Calculus

Example Question #71 : Exponential And Logarithmic Functions

Example Question #71 : Exponential And Logarithmic Functions

Example Question #11 : Solve Logarithmic Equations

Example Question #71 : Exponential And Logarithmic Functions

Example Question #81 : Exponential And Logarithmic Functions

Example Question #81 : Exponential And Logarithmic Functions

Example Question #14 : Solve Logarithmic Equations

Example Question #14 : Solve Logarithmic Equations

Example Question #18 : Solve Logarithmic Equations

All Precalculus Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: