Call Now to Set Up Tutoring:

(888) 888-0446

Precalculus : Exponential and Logarithmic Functions

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 #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

Example Question #21 : Solve Logarithmic Equations

Completely condense the logarithm: $\frac{1}{3}log_{10}x-\frac{1}{2}[log_{10}(x+y)]$ .

Possible Answers:

$log_{10}\frac{\sqrt[3]{x}}{\sqrt{xy}}$

$log_{10}\frac{\sqrt[3]{x}}{\sqrt{x+y}}$

None of the other answers.

$log_{10}\frac{\sqrt{x}}{\sqrt{x+y}}$

$log_{10}\frac{\sqrt{x}}{\sqrt{xy}}$

Correct answer:

$log_{10}\frac{\sqrt[3]{x}}{\sqrt{x+y}}$

Explanation:

$\frac{1}{3}log_{10}x-\frac{1}{2}[log_{10}(x+y)]$

Apply the Power property:

$log_{10}x^\frac{1}{3}-log_{10}(x+y)^\frac{1}{2}$

$log_{10}\sqrt[3]{x}-log_{10}\sqrt{x+y}$

Apply the quotient property:

$\mathbf{log_{10}\frac{\sqrt[3]{x}}{\sqrt{x+y}}}$

Report an Error

Example Question #21 : Solve Logarithmic Equations

Simplify

$\small \log \,3\,\,+\,\,\log\,7$

Possible Answers:

$\small \log\,\frac{3}{7}$

$\small \log\,37$

$\small \log\,21$

$\small \log\,10$

Correct answer:

$\small \log\,21$

Explanation:

By the property of the addition of logarithms with the same base

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

As such

$\small \small \log \,3\,\,+\,\,\log\,7=\log\,3\cdot7=\log\,21$

Report an Error

Example Question #23 : Solve Logarithmic Equations

Condense the following equation:

log(x)+4log(y)+log(3)-log(z)

Possible Answers:

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

Cannot simplify

$log(\frac{12xy}{z})$

log(12xyz)

log(x+4y+3-z)

Correct answer:

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

Explanation:

Let's use the properties of logarithms to condense this equation. We will use the follwing three properties

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

Product Rule log(x)+log(y)=log(xy)

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

Let's first use the power rule to rewrite the second term

log(x)+log(y^4)+log(3)-log(z)

Then, we'll use the product rule to combine the first threeterms

log(3xy^4)-log(z)

Lastly, we'll use the quotient rule to combine into one term

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

Report an Error

Example Question #22 : Solve Logarithmic Equations

Express the following in condensed form:

$\log(x-4)+\log(x)-\log{2}$

Possible Answers:

$\log\frac{x^2-4x}{2}$

$\log(x-4)+\log(x)-\log{2}$

$\log(x-4)+2$

$\log(x-4)(x)(2)$

Correct answer:

$\log\frac{x^2-4x}{2}$

Explanation:

To solve, simply remember that when you add logs, you can multiply their insides and when you subtract them, you can divide.

Thus,

$\\ \log(x-4)+\log(x)-\log{2}\\ \\=\log\frac{(x-4)x}{2}\\ \\=\log\frac{x^2-4x}{2}$

Report an Error

Example Question #23 : Solve Logarithmic Equations

Express the following in condensed form:

$\log(x^2)+\log(y^2)$

Possible Answers:

Same

$\log(x^2y^2)$

$\log(xy^4)$

$\log(x^2+y^2)$

Correct answer:

$\log(x^2y^2)$

Explanation:

To solve, simply remember that when add logs, you multiply their insides. Thus,

$\log{x^2}+\log{y^2}=\log(x^2y^2)$

Report an Error

View Pre-Calculus Tutors

Joseph
Certified Tutor

Bank Street College of Education , Master's/Graduate, Early Childhood and Childhood General Education.

View Pre-Calculus Tutors

Muhammad
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Mechanical Engineering.

View Pre-Calculus Tutors

Alejandro
Certified Tutor

Kennesaw State University, Bachelor of Science, Computer Software Engineering.

All Precalculus Resources

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

Popular Subjects

Spanish Tutors in Dallas Fort Worth, French Tutors in Miami, Physics Tutors in San Francisco-Bay Area, Math Tutors in New York City, GMAT Tutors in Houston, Chemistry Tutors in San Francisco-Bay Area, Math Tutors in Washington DC, SAT Tutors in Philadelphia, GRE Tutors in Seattle, Biology Tutors in Dallas Fort Worth

Popular Courses & Classes

LSAT Courses & Classes in Philadelphia, ISEE Courses & Classes in Dallas Fort Worth, Spanish Courses & Classes in Los Angeles, SAT Courses & Classes in Seattle, ISEE Courses & Classes in San Francisco-Bay Area, GMAT Courses & Classes in Dallas Fort Worth, ISEE Courses & Classes in Phoenix, Spanish Courses & Classes in Houston, ACT Courses & Classes in New York City, ACT Courses & Classes in San Diego

Popular Test Prep

ISEE Test Prep in Dallas Fort Worth, GMAT Test Prep in Dallas Fort Worth, GRE Test Prep in San Francisco-Bay Area, SSAT Test Prep in Washington DC, MCAT Test Prep in Atlanta, ACT Test Prep in San Diego, ACT Test Prep in Philadelphia, SAT Test Prep in Chicago, SAT Test Prep in Los Angeles, LSAT Test Prep in Phoenix

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 : Exponential and Logarithmic Functions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

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

Example Question #21 : Solve Logarithmic Equations

Example Question #21 : Solve Logarithmic Equations

Example Question #23 : Solve Logarithmic Equations

Example Question #22 : Solve Logarithmic Equations

Example Question #23 : Solve Logarithmic Equations

All Precalculus Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: