Write the single logarithm as a sum or difference of logs.log3(x9y2)\log_3\left(\frac{\sqrt{x}}{9y^2}\right)

12log⁡3x−2−2log⁡3y\frac12\log_3x-2-2\log_3y21log3x−2−2log3y

Write the single logarithm as a sum or difference of logs. log 3 ( x 9 y 2 ) \log_3\left(\frac{\sqrt{x}}{9y^2}\right)

In this problem, we're asked to write a single logarithm as a sum or difference of logs. And here, we're given log base 3 of the square root of x divided by 9 y squared, and we wanna go ahead and expand this out as much as possible. So let's go ahead and get started. Now the first thing I notice here is that these numbers are being divided, so I can go ahead and apply the quotient rule and change that division into the subtraction of 2 different logs. So if I expand this out, this becomes a log base 3 of the square root of x minus log base 3 of 9 y squared. Now that I've separated those 2 logs, I'm gonna go to each of these individual logs and see how else I can expand it. Now this log base 3 of the square root of x, I know that I can change the square root of x into an exponent to allow me to use the power rule. So this will become log base 3 of x to the power of onetwo. Now that I have that exponent, I can go ahead and apply the power rule and pull that exponent to the front of my log. So this becomes 1 half times log base 3 of x. And there is nothing else I can do to expand that log, so we are good. This is how this log is going to stay. Now let's go ahead and move on to our other log and focus on that. So looking over here, I have log base 3 of 9 y squared. But since 9 is just 3 squared and my y squared, I can go ahead and pull that squared out into an exponent. So this becomes a minus log base 3 of 3 y squared. And now that I have that exponent, I can again apply the power rule and pull that exponent to the front of my log. Now in doing that, this becomes 2 times log base 3 of 3 y, and that exponent goes away. Now I might want to stop here, but I can actually do even more to expand this because I see that this 3 and this y are multiplying each other. Now whenever we have things being multiplied, we can go ahead and apply the product rule and change that multiplication into the addition of 2 separate logs. Now this one half log base 3 of x is going to stay the same. We're still not worrying about that. Let's go ahead and separate this log base 3 of 3 y out. Now we want to make sure that we keep this negative 2 attached to it because it hasn't gone anywhere. So this is negative 2. And then separating this out using the product rule, this becomes log base 3 of 3 plus log base 3 of y. Now looking at this, I notice this log base 3 of 3. Since my base is the same as what I'm taking the log of, this really just becomes 1. So this is really minus 2 times 1 plus log base 3 of y. And now that I have gotten it to these multiple log expressions, I have simplified it so much, I can go ahead and distribute this 2 into it in order to get my final answer. So I'm gonna go ahead and pull this one half log base 3 of x down because even though we haven't worked with it for a while, it's still a part of our expression. So this is one half log base 3 of x and then a minus I'm gonna go ahead and distribute that to n. So minus 2 and then minus 2 log base 3 of y. Now this is my fully complete answer, and that is my sum or difference of logs that I have gotten from this original, much more compact log expression. Hopefully that makes sense. Let me know if you have questions.

Write the single logarithm as a sum or difference of logs.log⁡3(x9y2)\$$log_3$$\left(\frac{\sqrt{x}}{$$9y^2$$}\right)

12log⁡3x−2−2log⁡3y\frac12\log_3x-2-2\log_3y21log3x−2−2log3y

2log⁡3x−2−log⁡39y2\log_3x-2-\log_39y2log3x−2−log39y

12log⁡3x+2log⁡33y\frac12\log_3x+2\log_33y21log3x+2log33y

12log⁡3x−2log⁡39y\frac12\log_3x-2\log_39y21log3x−2log39y

Table of contents

Skip topic navigation

0. Review4h 36m

Algebraic Expressions
37m

Exponents
32m

Polynomials Intro
19m

Multiplying Polynomials
36m

Factoring Polynomials
1h 2m

Radical Expressions
15m

Simplifying Radical Expressions
35m

Rationalize Denominator
15m

Rational Exponents
4m

Pythagorean Theorem & Basics of Triangles
17m

1. Equations & Inequalities3h 26m

Linear Equations
31m

Rational Equations
21m

The Imaginary Unit
6m

Powers of i
11m

Complex Numbers
41m

Intro to Quadratic Equations
18m

The Square Root Property
11m

Completing the Square
12m

The Quadratic Formula
19m

Choosing a Method to Solve Quadratics
9m

Linear Inequalities
22m

2. Graphs & Functions4h 0m

Graphs and Coordinates
7m

Two-Variable Equations
23m

Lines
1h 12m

Intro To Functions And Graphs
35m

Common Functions
5m

Transformations
45m

Function Operations
21m

Function Composition
29m

3. Polynomial and Rational Functions3h 18m

Quadratic Functions
49m

Understanding Polynomial Functions
34m

Graphing Polynomial Functions
30m

Intro to Rational Functions
9m

Asymptotes
35m

Graphing Rational Functions
38m

4. Inverse, Exponential, and Logarithmic Functions2h 28m

Intro to Exponential Functions
9m

Graphing Exponential Functions
25m

The Number e
8m

Intro to Logarithms
22m

Graphing Logarithmic Functions
18m

Properties of Logarithms
27m

Solving Exponential & Logarithmic Equations
35m

5. Trigonometric Functions5h 25m

Angles in Standard Position
12m

Coterminal Angles
8m

Complementary & Supplementary Angles
10m

Converting between Degrees & Radians
8m

Trigonometric Functions on Right Triangles
45m

Special Right Triangles
30m

Cofunctions of Complementary Angles
26m

Solving Right Triangles
23m

Defining the Unit Circle
14m

Trig Functions on the Unit Circle
9m

Common Values of Sine, Cosine, & Tangent
11m

Reference Angles
38m

Reciprocal Trig Functions on the Unit Circle
6m

Graphs of the Sine and Cosine Functions
32m

Phase Shifts
14m

Graphs of Secant and Cosecant Functions
10m

Graphs of Tangent and Cotangent Functions
21m

6. Analytic Trigonometry4h 15m

Inverse Sine, Cosine, & Tangent
28m

Evaluate Composite Trig Functions
44m

Linear Trigonometric Equations
28m

Introduction to Trig Identities
1h 4m

Sum and Difference Identities
1h 3m

Double Angle Identities
16m

Solving Trig Equations Using Identities
9m

7. Applications of Trigonometry1h 38m

The Law of Sines
49m

The Law of Cosines
30m

Area of SAS & ASA Triangles
19m

8. Polar Coordinates and Complex Numbers4h 20m

Polar Coordinate System
29m

Convert Points Between Polar & Rectangular Coordinates
26m

Convert Equations Between Polar & Rectangular Forms
29m

Graphing Other Common Polar Equations
39m

Graphing Parametric Equations
12m

Eliminate the Parameter
32m

Writing Parametric Equations
21m

Graphing Complex Numbers
6m

Polar Form of Complex Numbers
22m

Products and Quotients of Complex Numbers
14m

Powers of Complex Numbers (DeMoivre's Theorem)
23m

9. Vectors2h 25m

Geometric Vectors
28m

Vectors in Component Form
32m

Direction of a Vector
23m

Unit Vectors and i & j Notation
27m

Dot Product
22m

Cross Product
11m

10. Systems of Equations & Matrices4h 5m

Two Variable Systems of Linear Equations
1h 7m

Introduction to Matrices
1h 11m

Determinants and Cramer's Rule
1h 3m

Graphing Systems of Inequalities
42m

11. Analytic Geometry2h 36m

Introduction to Conic Sections
6m

Circles
26m

Ellipses: Standard Form
33m

Parabolas
36m

Hyperbolas at the Origin
40m

Hyperbolas NOT at the Origin
12m

12. Sequences, Series, and Induction1h 22m

Sequences
40m

Arithmetic Sequences
25m

Geometric Sequences
15m

13. Combinatorics and Probability1h 45m

Factorials
11m

Combinatorics
46m

Probability
47m

4. Inverse, Exponential, and Logarithmic Functions

Properties of Logarithms

Algebra & Trigonometry

4. Inverse, Exponential, and Logarithmic Functions

Properties of Logarithms

Previous problemNext problem

Struggling with Algebra & Trigonometry?

Join thousands of students who trust us to help them ace their exams!

Multiple Choice

Write the single logarithm as a sum or difference of logs.
$\log_{3} (\frac{\sqrt{x}}{9 y^{2}})$

$2$\log$_3x-2-$\log$_39y$

$$\frac$12$\log$_3x-2-2$\log$_3y$

$$\frac$12$\log$_3x+2$\log$_33y$

$$\frac$12$\log$_3x-2$\log$_39y$

0 Comments

Previous problemNext problem

5:36m

Watch next

Master Change of Base Property with a bite sized video explanation from Callie Rethman

Struggling with Algebra & Trigonometry?

Write the single logarithm as a sum or difference of logs.log3(x9y2)\(\log\)_3\(\left\)(\(\frac{\sqrt{x}\)}{9y^2}\(\right\))

Watch next

Write the single logarithm as a sum or difference of logs.
$\log_{3} (\frac{\sqrt{x}}{9 y^{2}})$