We are open Saturday and Sunday!

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 #1 : Solve Logarithmic Equations

Evaluate a logarithm.

What is $log_{10}1000$ ?

Possible Answers:

$\frac{1}{3}$

100

Correct answer:

Explanation:

The derifintion of logarithm is:

$log_{a}y=x \Leftrightarrow a^{x}=y.$

In this problem,

a= 10, y=1000

$10^{3}=1000$

Therefore, x=3

Report an Error

Example Question #1 : Solve Logarithmic Equations

Solve for in the following logarithmic equation:

$log_{10} x^2 - log_{10} 2x = 2$

Possible Answers:

x = 10

x = 1000

None of the other choices

x=100

x = 0

Correct answer:

None of the other choices

Explanation:

Using the rules of logarithms,

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

Hence,

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

So exponentiate both sides with a base 10:

$10^{log_{10} \frac{x}{2}} = 10^{2}$

The exponent and the logarithm cancel out, leaving:

$\frac{x}{2}=10^2$

$x=100\cdot 2=200$

This answer does not match any of the answer choices, therefore the answer is 'None of the other choices'.

Report an Error

Example Question #1 : Solve Logarithmic Equations

Solve the following logarithmic equation:

$2\log _3(x)=\log _3(4)+\log _3(x-1)$

Possible Answers:

x=1

x=2

x=-1

x=-4

x=4

Correct answer:

x=2

Explanation:

In order to solve this equation, we must apply several properties of logarithms. First we notice the term on the left side of the equation, which we can rewrite using the following property:

$a\log_b (x)=\log_b (x^a)$

Where a is the coefficient of the logarithm and b is some arbitrary base. Next we look at the right side of the equation, which we can rewrite using the following property for the addition of logarithms:

$\log _ba+\log _bc=\log_b (a\cdot c)$

Using both of these properties, we can rewrite the logarithmic equation as follows:

$2\log _3(x)=\log _3(4)+\log _3(x-1)$

$\log _3(x^2)=\log _3(4(x-1))$

We have the same value for the base of the logarithm on each side, so the equation then simplifies to the following:

$x^2=4x-4\rightarrow x^2-4x+4=0$

Which we can then factor to solve for :

$(x-2)(x-2)=0\rightarrow x=2$

Report an Error

Example Question #2 : Solve Logarithmic Equations

Solve the equation for .

$\ln\frac{cx}{ab+c} = b+c$

Possible Answers:

$x = \frac{e^a}{e^{bc}}}(ab+c)$

$x = \frac{e^be^c(ab+c)}{c}}$

None of the other answers.

ab-e^c

$x = \frac{e^b}{c(ab+c)e^c}$

Correct answer:

$x = \frac{e^be^c(ab+c)}{c}}$

Explanation:

We solve the equation as follows:

$\ln \frac{cx}{ab+c} = b+c$

Exponentiate both sides.

$\frac{cx}{ab+c}= e^{b+c}$

Apply the power rule on the right hand side.

$\frac{cx}{ab+c} = e^be^c$

Multiply by ab+c .

cx = e^be^c(ab+c)

Divide by .

$x =\frac{e^be^c(ab+c)}{c}$

Report an Error

Example Question #4 : Solve Logarithmic Equations

Solve for :

log(27)- 2log(x)=log(x)

Possible Answers:

x=54

x=3

x=-3

$x=-\frac{1}{27}$

x=3.67

Correct answer:

x=3

Explanation:

First, simplify the logarithmic expressions on the left side of the equation:

2log(x) can be re-written as log(x^2) .

Now we have:

log(27)-log(x^2)=log(x) .

The left can be consolidated into one log expression using the subtraction rule:

$log\left(\frac{27}{x^2}\right)=log(x)$ .

We now have log on both sides, so we can be confident that whatever is inside these functions is equal:

$\frac{27}{x^2}=x$ to continue solving, multiply by x^2 on both sides:

27 = x^3 take the cube root:

3 = x

Report an Error

Example Question #1 : Solve Logarithmic Equations

$(\ln (x^2))^2=\ln (x^4)+3$ .

Solve for .

Possible Answers:

$e^{4/3}$

$e^{-3/2}$

$e^{2/3}$

$e^{-1/2}$

$e^{3/2}$

Correct answer:

$e^{3/2}$

Explanation:

First bring the inside exponent in front of the natural log.

$(\ln (x^2))^2=\ln (x^4)+3\rightarrow(2\ln (x))^2=4\ln (x)+3$ .

Next simplify the first term and bring all the terms on one side of the equation.

$4(\ln (x))^2-4\ln (x)-3=0$ .

Next, let set

$\ln (x)=A$ , so 4A^2-4A-3=0 .

Now use the quadratic formula to solve for .

$\\ \frac{-b\pm \sqrt{b^2-4(a)(c)}}{2a}\\ \\=\frac{4\pm\sqrt{16-4(4)(-3)}}{2(4)}\\ \\=\frac{4\pm \sqrt{16+48}}{8}\\ \\=4\pm \frac{\sqrt{64}}{8}\\ \\=\frac{4\pm 8}{8}\\ \\=\frac{3}{2}; -\frac{1}{2}$

and thus, $A=-\frac{1}{2}$ and $A=\frac{3}{2}$ .

Now substitute with $\ln (x)$ .

So, $\ln (x)=-\frac{1}{2}\rightarrow \O$ since $x\nleqslant0$ and $\ln (x)=\frac{3}{2}\rightarrow x=e^{3/2}$ .

Thus, $x=e^{3/2}$ .

Report an Error

Example Question #6 : Solve Logarithmic Equations

Solve the logarithmic equation: $\ln\sqrt{x+2}=2$

Possible Answers:

x=e^4-4

x=e^2+4

None of the other answers.

x=e^2-2

x=e^4-2

Correct answer:

x=e^4-2

Explanation:

$\ln\sqrt{x+2}=2$

Exponentiate each side to cancel the natural log:

$e^{\ln\sqrt{x+2}}=e^2$

$\sqrt{x+2}=e^2$

Square both sides:

$(\sqrt{x+2})^2=(e^2)^2$

x+2=e^4

Isolate x:

x=e^4-2

Report an Error

Example Question #1166 : Pre Calculus

Solve for x: $\log(2x+1) = 3$

Possible Answers:

499.5

9.5

Correct answer:

499.5

Explanation:

The base of a logarithm is 10 by default:

$\log _{10 } (2 x + 1 ) = 3$ convert to exponent to isolate x

10^3 = 2x + 1

1,000 = 2x + 1 subtract 1 from both sides

999 = 2x divide both sides by 2

499.5 = x

Report an Error

Example Question #1167 : Pre Calculus

Solve for x: $\log _3 x - \log _3 7 = 2$

Possible Answers:

Correct answer:

Explanation:

First, condense the left side into one logarithm:

$\log _3 \frac{x}{7 } = 2$ convert to an exponent

$3^2 = \frac{x}{7 }$

$9 = \frac{x}{7}$ multiply both sides by 7

63 = x

Report an Error

Example Question #1168 : Pre Calculus

Solve for x: $\log _4 x + \log _ 4 (3x - 7 ) = 3$

Possible Answers:

3.95, - 2.40

5.93 , -3.60

no solution

2.5

8.90, - 5.40

Correct answer:

5.93 , -3.60

Explanation:

First, consolidate the left side into one logarithm:

$\log _4 (3x^2 - 7x ) = 3$ convert to an exponent

3x^2 - 7x = 64 subtract 64 from both sides

3x^2 - 7 x - 64 = 0 now we can solve using the quadratic formula:

$x = \frac{7 \pm \sqrt{49 - 4(3)(-64)}}{6 } \approx 5.93, -3.60$

Report an Error

View Pre-Calculus Tutors

Dalton
Certified Tutor

University of Massachusetts Amherst, Doctor of Philosophy, Mathematics.

View Pre-Calculus Tutors

John
Certified Tutor

Arizona State University, Bachelor of Science, Microbiology.

View Pre-Calculus Tutors

Vandana
Certified Tutor

Jawaharlal Nehru Technological University, Bachelor of Technology, Civil Engineering. Florida International University, Maste...

All Precalculus Resources

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

Popular Subjects

SSAT Tutors in Washington DC, Physics Tutors in Atlanta, Chemistry Tutors in Atlanta, Statistics Tutors in Atlanta, Math Tutors in Dallas Fort Worth, Spanish Tutors in Los Angeles, Biology Tutors in San Francisco-Bay Area, French Tutors in New York City, SSAT Tutors in Atlanta, English Tutors in Los Angeles

Popular Courses & Classes

ISEE Courses & Classes in Washington DC, ACT Courses & Classes in New York City, SAT Courses & Classes in Denver, LSAT Courses & Classes in Atlanta, MCAT Courses & Classes in San Diego, ISEE Courses & Classes in Dallas Fort Worth, MCAT Courses & Classes in Seattle, ISEE Courses & Classes in Atlanta, ISEE Courses & Classes in San Diego, Spanish Courses & Classes in Boston

Popular Test Prep

LSAT Test Prep in New York City, LSAT Test Prep in Atlanta, ACT Test Prep in Miami, MCAT Test Prep in Atlanta, MCAT Test Prep in Denver, LSAT Test Prep in Houston, LSAT Test Prep in San Francisco-Bay Area, ISEE Test Prep in San Diego, GRE Test Prep in Los Angeles, ISEE Test Prep in New York City

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 #1 : Solve Logarithmic Equations

Example Question #1 : Solve Logarithmic Equations

Example Question #1 : Solve Logarithmic Equations

Example Question #2 : Solve Logarithmic Equations

Example Question #4 : Solve Logarithmic Equations

Example Question #1 : Solve Logarithmic Equations

Example Question #6 : Solve Logarithmic Equations

Example Question #1166 : Pre Calculus

Example Question #1167 : Pre Calculus

Example Question #1168 : Pre Calculus

All Precalculus Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: