We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Precalculus : Pre-Calculus

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 #2 : Exponential Equations And Inequalities

Solving an exponential equation.

Solve

$10^{3x}=100$ $\displaystyle 10^{3x}=100$

Possible Answers:

$x=\frac{2}{3}$ $\displaystyle x=\frac{2}{3}$

x=10 $\displaystyle x=10$

$x=\frac{3}{2}$ $\displaystyle x=\frac{3}{2}$

x=2 $\displaystyle x=2$

x=3 $\displaystyle x=3$

Correct answer:

$x=\frac{2}{3}$ $\displaystyle x=\frac{2}{3}$

Explanation:

Use log $\displaystyle log$ (which is just $log_{10}$ $\displaystyle log_{10}$, by convention) to solve.

$log(10^{3x})=log(100)$ $\displaystyle log(10^{3x})=log(100)$

$log(10^{3x})=log(10^2)$ $\displaystyle log(10^{3x})=log(10^2)$

3x=2 $\displaystyle 3x=2$

$x=\frac{2}{3}$ $\displaystyle x=\frac{2}{3}$.

Report an Error

Example Question #1 : Exponential Equations And Inequalities

Solve the equation for $\displaystyle x$ using the rules of logarithms.

$\ln(17x)+\ln\left(\frac{x}{3}\right)+\log_3(x^2)=4$ $\displaystyle \ln(17x)+\ln\left(\frac{x}{3}\right)+\log_3(x^2)=4$

Possible Answers:

$x\approx 75.15$ $\displaystyle x\approx 75.15$

x=3 $\displaystyle x=3$

$x\approx 1.04$ $\displaystyle x\approx 1.04$

x=9 $\displaystyle x=9$

$x\approx1.86$ $\displaystyle x\approx1.86$

Correct answer:

$x\approx 1.04$ $\displaystyle x\approx 1.04$

Explanation:

Expanding the logarithms into sums of logarithms will cancel out the first two x terms, resulting in the equation:

$\ln(17)+ln(3)+2\log_3(x)=4$ $\displaystyle \ln(17)+ln(3)+2\log_3(x)=4$

Combining the first and second terms, then subtracting the new term over will allow you to isolate the variable term.

Divide both sides of the equation by 2, then exponentiate with 3.

Evaluating this term numerically will give the correct answer.

$2\log_3(x)=4-\ln(51)\implies \log_3(x)=2-\frac{\ln(51)}{2}\\ \\ \implies x=3^{2-\ln(51)/2}\approx1.04$ $\displaystyle 2\log_3(x)=4-\ln(51)\implies \log_3(x)=2-\frac{\ln(51)}{2}\\ \\ \implies x=3^{2-\ln(51)/2}\approx1.04$

Report an Error

Example Question #2 : Exponential Equations And Inequalities

Solve the following equation:

$\displaystyle 10^x=175$

Possible Answers:

$x\approx0.224$ $\displaystyle x\approx0.224$

$x\approx4.48$ $\displaystyle x\approx4.48$

$x\approx22.4$ $\displaystyle x\approx22.4$

x=2.24 $\displaystyle x=2.24$

$x\approx2.24$ $\displaystyle x\approx2.24$

Correct answer:

$x\approx2.24$ $\displaystyle x\approx2.24$

Explanation:

To solve this equation, recall the following property:

$\displaystyle log_by=x$ Can be rewritten as b^x=y $\displaystyle b^x=y$

$\displaystyle 10^x=175$

$log_{10}175=x$ $\displaystyle log_{10}175=x$

Evaluate with your calculator to get

$x\approx2.24$ $\displaystyle x\approx2.24$

Report an Error

Example Question #3 : Use Logarithms To Solve Exponential Equations And Inequalities

Solve

$\frac{e^{x^2-2}}{e^{3x}}=e^{2x-6}$ $\displaystyle \frac{e^{x^2-2}}{e^{3x}}=e^{2x-6}$.

Possible Answers:

$\displaystyle x=1$ or $4$

x=4 $\displaystyle x=4$

x=1 $\displaystyle x=1$

$\displaystyle x=0$ or 1$

$\displaystyle x=7$ or $4$

Correct answer:

$\displaystyle x=1$ or $4$

Explanation:

After using the division rule to simplify the left hand side you can take the natural log of both sides.

If you then combine like terms you get a quadratic equation which factors to,

$\displaystyle (x-1)(x-4)=0$.

Setting each binomial equal to zero and solving for $\displaystyle x$ we get the solution to be $\displaystyle x=1$ or $4$.

Report an Error

Example Question #1 : Use Logarithms To Solve Exponential Equations And Inequalities

Solve for x: 3^x=243 $\displaystyle 3^x=243$

Possible Answers:

x=3 $\displaystyle x=3$

x=1 $\displaystyle x=1$

x=5 $\displaystyle x=5$

x=-1 $\displaystyle x=-1$

x=4 $\displaystyle x=4$

Correct answer:

x=4 $\displaystyle x=4$

Explanation:

$3^x=3^4\rightarrow x=4$ $\displaystyle 3^x=3^4\rightarrow x=4$

Report an Error

Example Question #4 : Use Logarithms To Solve Exponential Equations And Inequalities

Solve for x in the following equation: $3^{5x-3}=9^{2x}$ $\displaystyle 3^{5x-3}=9^{2x}$

Possible Answers:

x=0 $\displaystyle x=0$

x=1 $\displaystyle x=1$

x=-1 $\displaystyle x=-1$

x=5 $\displaystyle x=5$

x=3 $\displaystyle x=3$

Correct answer:

x=3 $\displaystyle x=3$

Explanation:

$3^{5x-3}=3^{(2)(2x)}\rightarrow 5x-3=4x\rightarrow x=3$ $\displaystyle 3^{5x-3}=3^{(2)(2x)}\rightarrow 5x-3=4x\rightarrow x=3$

Report an Error

Example Question #4 : Use Logarithms To Solve Exponential Equations And Inequalities

Solve for x using the rules of logarithms: $\log_{3}2x-\log_{3}(x-3)=1$ $\displaystyle \log_{3}2x-\log_{3}(x-3)=1$

Possible Answers:

x=3 $\displaystyle x=3$

x=1 $\displaystyle x=1$

x=4 $\displaystyle x=4$

x=2 $\displaystyle x=2$

x=9 $\displaystyle x=9$

Correct answer:

x=9 $\displaystyle x=9$

Explanation:

$log_3 \frac{2x}{x-3}=1\rightarrow 3^{log_3 \frac{2x}{x-3}}=3^1\rightarrow \frac{2x}{x-3}=3\rightarrow x=9$ $\displaystyle log_3 \frac{2x}{x-3}=1\rightarrow 3^{log_3 \frac{2x}{x-3}}=3^1\rightarrow \frac{2x}{x-3}=3\rightarrow x=9$

Report an Error

Example Question #5 : Use Logarithms To Solve Exponential Equations And Inequalities

Solve for x: $\log_{4}(3x-2)=2$ $\displaystyle \log_{4}(3x-2)=2$

Possible Answers:

x=4 $\displaystyle x=4$

x=3 $\displaystyle x=3$

x=6 $\displaystyle x=6$

x=2 $\displaystyle x=2$

x=5 $\displaystyle x=5$

Correct answer:

x=6 $\displaystyle x=6$

Explanation:

$log_4 (3x-2)=2\rightarrow 3x-2=4^2\rightarrow 3x=18\rightarrow x=6$ $\displaystyle log_4 (3x-2)=2\rightarrow 3x-2=4^2\rightarrow 3x=18\rightarrow x=6$

Report an Error

Example Question #7 : Use Logarithms To Solve Exponential Equations And Inequalities

Simplify the log expression: $\displaystyle logx*log2$

Possible Answers:

2logx $\displaystyle 2logx$

Cannot be simplified any further

$\displaystyle log(x)^2$

log2x $\displaystyle log2x$

2log2x $\displaystyle 2log2x$

Correct answer:

Cannot be simplified any further

Explanation:

The logarithmic expression is as simplified as can be.

Report an Error

Example Question #1231 : Pre Calculus

Which of the following could NOT be a possible number of solutions of a system of quadratic and linear equations?

Possible Answers:

How many times the graphs intersect

Correct answer:

Explanation:

Recall that the solution of a system of equations is given by the intersection points of the graphs. Thus, this question is really asking how many times a parabola and a line can intersect. Visualize a parabola and a line. For this purpose, let's say that the parabola is facing up. If a line is drawn horizontally under the vertex, then it would not intersect the parabola at all, so the system would have no solutions. If the line is tangent to, or just skims the edge, of the parabola then it would only intersect once and the system would have one solution. If the line goes straight through the parabola, then it would intersect twice. There is no other option for the orientation of the line and parabola. Thus, there cannot be three solutions.

Report an Error

View Pre-Calculus Tutors

Kinza
Certified Tutor

University of Georgia, Bachelor of Science, Biology, General.

View Pre-Calculus Tutors

Tara
Certified Tutor

Montclair State University, Bachelor of Science, Mathematics. Montclair State University, Master of Arts, Educational Adminis...

View Pre-Calculus Tutors

Chernyeh
Certified Tutor

Purdue University-Main Campus, Bachelor of Science, Business Administration and Management.

All Precalculus Resources

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

Popular Subjects

English Tutors in Chicago, SSAT Tutors in Seattle, Spanish Tutors in Washington DC, English Tutors in Atlanta, GRE Tutors in Chicago, Spanish Tutors in Seattle, Computer Science Tutors in Atlanta, SSAT Tutors in New York City, GRE Tutors in Washington DC, Chemistry Tutors in Chicago

Popular Courses & Classes

SSAT Courses & Classes in San Diego, SAT Courses & Classes in Boston, ACT Courses & Classes in Philadelphia, SSAT Courses & Classes in Atlanta, ACT Courses & Classes in Washington DC, LSAT Courses & Classes in San Francisco-Bay Area, SAT Courses & Classes in Seattle, ACT Courses & Classes in Seattle, GMAT Courses & Classes in San Diego, ISEE Courses & Classes in Los Angeles

Popular Test Prep

MCAT Test Prep in Phoenix, LSAT Test Prep in Atlanta, ACT Test Prep in Seattle, GRE Test Prep in New York City, SAT Test Prep in San Diego, SSAT Test Prep in Atlanta, GMAT Test Prep in Los Angeles, ACT Test Prep in Philadelphia, ACT Test Prep in Houston, MCAT Test Prep in Chicago

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 : Pre-Calculus

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #2 : Exponential Equations And Inequalities

Example Question #1 : Exponential Equations And Inequalities

Example Question #2 : Exponential Equations And Inequalities

Example Question #3 : Use Logarithms To Solve Exponential Equations And Inequalities

Example Question #1 : Use Logarithms To Solve Exponential Equations And Inequalities

Example Question #4 : Use Logarithms To Solve Exponential Equations And Inequalities

Example Question #4 : Use Logarithms To Solve Exponential Equations And Inequalities

Example Question #5 : Use Logarithms To Solve Exponential Equations And Inequalities

Example Question #7 : Use Logarithms To Solve Exponential Equations And Inequalities

Example Question #1231 : Pre Calculus

All Precalculus Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: