Rewrite the log expression as the sum or difference of multiple logs.log10(8x35y)\log_{10}\left(\frac{8x^3}{5y}\right)

3log⁡102+3log⁡10x−log⁡105−log⁡10y3\$$log_{10}2+3$$\$$log_{10}x$$-\$$log_{10}5$$-\$$log_{10}y3log10$$2+3log10x−log105−log10y

Rewrite the log expression as the sum or difference of multiple logs. log 10 ( 8 x 3 5 y ) \log_{10}\left(\frac{8x^3}{5y}\right)

Here we're asked to rewrite the log expression as the sum or difference of multiple logs. And we're given here log base 10 of eight x cubed over five y. So we wanna expand this as much as possible. Looking at the argument of this log, I see that these two things are being divided. So I'm gonna start by applying my quotient property to rewrite this as the difference of logs. So this is gonna be a log base 10 of eight x cubed, taking that first argument, and then minus log base 10 of a five y. Now we're not done yet at all because there's still a lot we can do to simplify. Looking at this first log expression, I have eight multiplying x cubed. So I can rewrite that as a log base 10 of eight plus log base 10 of x cubed, applying the product property. Now looking at this second log expression here, I have five multiplying y. So I can again, apply my product property here, making sure to enclose this first in parentheses so that I'm capturing the fact that this is negative out front. So rewriting this as log base 10 of five plus log base 10 of y applying that product property. Now I'm going to go ahead and distribute that negative here. So that's going to make this all a log base 10 of eight plus log base 10 of X cubed minus log base 10 of five minus a log base 10 of y, just making both of those terms negative. Now there's still more I can do here because we still have our power property to apply and I can actually apply my power property to two of these logs. Now obviously with this log base 10 of x cubed, I have this power already there. I can pull that out front and multiply by it, but there's another place I can do this as well. Because I can rewrite eight as an exponential expression, as two cubed, I can also pull this power out front in order to further expand this out. So I'm gonna write this as three times a log base 10 of two plus three times a log base 10 of x and then minus log base 10 of five minus log base 10 of y. So now we have expanded that as much as we possibly can and we actually applied all of our log properties in this problem. So if you think you're done after one step, be sure to double check and make sure there are no other opportunities to expand. I'll see you in the next one.

Rewrite the log expression as the sum or difference of multiple logs.log⁡10(8x35y)\log_{10}\left(\frac{$$8x^3$$}{5y}\right)

3log⁡102+3log⁡10x−log⁡105−log⁡10y3\$$log_{10}2+3$$\$$log_{10}x$$-\$$log_{10}5$$-\$$log_{10}y3log10$$2+3log10x−log105−log10y

3log⁡102+3log⁡10x−log⁡105+log⁡10y3\$$log_{10}2+3$$\$$log_{10}x$$-\$$log_{10}5$$+\$$log_{10}y$$

−3log⁡102+3log⁡10x−log⁡10y-3\$$log_{10}2+3$$\$$log_{10}x$$-\$$log_{10}y$$

−3log⁡102+3log⁡10x+log⁡10y-3\$$log_{10}2+3$$\$$log_{10}x$$+\$$log_{10}y$$−3log102+3log10x+log10y

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1. Review of Real Numbers2h 43m

Simplifying Fractions
30m

Evaluating Exponents
29m

Evaluating Expressions
15m

Translating Phrases to Expressions
16m

Translating Sentences to Equations
9m

The Distributive Property
17m

Simplifying Expressions
44m

2. Linear Equations and Inequalities5h 35m

Solving Linear Equations
2h 23m

Linear Inequalities in One Variable
46m

Set Operations and Compound Inequalities
1h 7m

Absolute Value Equations
38m

Absolute Value Inequalities
39m

3. Solving Word Problems2h 46m

Introduction to Problem Solving
37m

Formulas
25m

Percent Problem Solving
59m

Mixture Problem Solving
43m

4. Graphs and Functions5h 12m

The Rectangular Coordinate System
44m

Graph Linear Equations in Two Variables
24m

Graph Linear Equations Using Intercepts
23m

Slope of a Line
44m

Slope-Intercept Form
38m

Point Slope Form
22m

Linear Inequalities in Two Variables
28m

Introduction to Relations and Functions
53m

Function Notation
15m

Composition of Functions
17m

5. Systems of Linear Equations1h 53m

Solving Systems of Linear Equations by Graphing
29m

Solving Systems of Linear Equations by Substitution
15m

Solving Systems of Linear Equations by Elimination
45m

Systems of Linear Inequalities
23m

6. Exponents, Polynomials, and Polynomial Functions3h 17m

The Product Rule
9m

The Quotient Rule
10m

Intro to the Power Rules
17m

The Power of a Quotient Rule
13m

Negative Exponents
24m

Simplifying Exponential Expressions Using All Exponent Rules
6m

Intro to Polynomials
21m

Adding and Subtracting Polynomials
20m

Multiplying Polynomials
38m

Special Products
34m

7. Factoring2h 49m

Factoring by Greatest Common Factor & Grouping
37m

Factoring Trinomials of the Form x² + bx + c
11m

Factoring Trinomials of the Form ax² + bx + c
37m

Factoring Special Products
41m

Quadratic Equations & Applications
41m

8. Rational Expressions and Functions3h 44m

Simplifying Rational Expressions
42m

Multiplying and Dividing Rational Expressions
25m

Adding and Subtracting Rational Expressions with Common Denominators
19m

Least Common Denominators
32m

Adding and Subtracting Rational Expressions with Different Denominators
32m

Rational Equations
44m

Direct & Inverse Variation
27m

9. Roots, Radicals, and Complex Numbers3h 56m

Introduction to Radical Expressions
46m

Simplifying Radical Expressions
47m

Adding and Subtracting Radicals
19m

Multiplying, Dividing, and Rationalizing Radicals
53m

Introduction to Complex Numbers
18m

Multiplying and Dividing Complex Numbers
51m

10. Quadratic Equations and Functions3h 1m

The Square Root Property
25m

Completing the Square
26m

The Quadratic Formula
40m

Graphing Quadratic Equations
1h 28m

11. Inverse, Exponential, & Logarithmic Functions2h 31m

Introduction to Inverse Functions
25m

Exponential Functions
35m

Introduction to Logarithmic Functions
33m

Properties of Logarithms
55m

12. Conic Sections & Systems of Nonlinear Equations58m

Graphing Circles
32m

Ellipses
15m

Parabolas
10m

13. Sequences, Series, and the Binomial Theorem1h 51m

Introduction to Sequences
40m

Arithmetic Sequences
27m

Geometric Sequences
29m

Factorials
13m

11. Inverse, Exponential, & Logarithmic Functions

Properties of Logarithms

Intermediate Algebra

11. Inverse, Exponential, & Logarithmic Functions

Properties of Logarithms

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Multiple Choice

Rewrite the log expression as the sum or difference of multiple logs.
$\log_{10} (\frac{8 x^{3}}{5 y})$

$3 \log_{10} 2 + 3 \log_{10} x - \log_{10} 5 + \log_{10} y$

$- 3 \log_{10} 2 + 3 \log_{10} x - \log_{10} y$

$3$\log$_{10}2+3$\log$_{10}x-$\log$_{10}5-$\log$_{10}y$

$-3$\log$_{10}2+3$\log$_{10}x+$\log$_{10}y$

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Verified step by step guidance

Start with the given logarithmic expression: $\log_{10}\left(\frac{8x^3}{5y}\right)$.

Use the logarithm property for division: $\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N$. So rewrite the expression as $\log_{10}(8x^3) - \log_{10}(5y)$.

Apply the logarithm property for multiplication: $\log_b(MN) = \log_b M + \log_b N$. This breaks down each log term into sums: $\log_{10} 8 + \log_{10} x^3 - (\log_{10} 5 + \log_{10} y)$.

Use the power rule of logarithms: $\log_b(M^k) = k \log_b M$. Apply this to $\log_{10} x^3$ to get $3 \log_{10} x$.

Express 8 as \$2^3$ and apply the power rule again: $\log_{10} 8 = \log_{10} (2^3) = 3 \log_{10} 2$. Now combine all parts to write the expression as $3 \log_{10} 2 + 3 \log_{10} x - \log_{10} 5 - \log_{10} y$.

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Master Power Property of Logs Example 2 with a bite sized video explanation from Patrick Ford

Struggling with Intermediate Algebra?

Rewrite the log expression as the sum or difference of multiple logs.log10(8x35y)\(\log\)_{10}\(\left\)(\(\frac{8x^3}{5y}\]\right\))

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Rewrite the log expression as the sum or difference of multiple logs.
$\log_{10} (\frac{8 x^{3}}{5 y})$