/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 9 Write the expression as one loga... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 9

Write the expression as one logarithm. (a) \(\log _{3} x+\log _{3}(5 y)\) (b) \(\log _{3}(2 z)-\log _{3} x\) (c) \(\frac{1}{5} \log _{3} y\)

Short Answer

Expert verified
(a) \(\log_{3} (5xy)\); (b) \(\log_{3} \left(\frac{2z}{x}\right)\); (c) \(\log_{3} (y^{1/5})\)."

Step by step solution

01

Applying the Product Rule

The expression given is \(\log_{3} x + \log_{3} (5y)\). The Product Rule of logarithms states that \(\log_{b} a + \log_{b} c = \log_{b} (ac)\). Apply this to combine the logarithms: \(\log_{3} (x \cdot 5y)\). Simplify the expression to obtain \(\log_{3} (5xy)\).
02

Applying the Quotient Rule

The expression given is \(\log_{3} (2z) - \log_{3} x\). The Quotient Rule of logarithms indicates that \(\log_{b} a - \log_{b} c = \log_{b} \left(\frac{a}{c}\right)\). Apply this rule to the expression: \(\log_{3} \left(\frac{2z}{x}\right)\). This combines the two logarithms into one.
03

Applying the Power Rule

The expression given is \(\frac{1}{5} \log_{3} y\). The Power Rule of logarithms states that \(c \log_{b} a = \log_{b} (a^c)\). Apply this rule by rewriting the expression as \(\log_{3} (y^{1/5})\). This represents the expression as one logarithm.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Product Rule of Logarithms
When you're dealing with logarithms, one of the most handy tools in your mathematical toolkit is the product rule. This rule makes it super easy to combine two logarithms into a single one. The magic formula is:
  • \(\log_{b} a + \log_{b} c = \log_{b} (ac)\)
This means if you're adding two logs with the same base, you can multiply the inside parts of the logs together.
For example, let's consider the problem: \(\log_{3} x + \log_{3} (5y)\). When we apply the product rule here, we multiply \(x\) and \(5y\) together to get \(5xy\).
So, \(\log_{3} x + \log_{3} (5y)\) becomes \(\log_{3} (5xy)\), combining the two logarithms into just one. This step simplifies expressions, making them much easier to handle!
Quotient Rule of Logarithms
Another powerful tool for simplifying logarithms is the quotient rule. This rule helps us combine logs with a subtraction sign between them into a single expression. It goes like this:
  • \(\log_{b} a - \log_{b} c = \log_{b} \left(\frac{a}{c}\right)\)
The minus sign in front of the second log indicates a division.
For instance, consider the expression \(\log_{3} (2z) - \log_{3} x\). By using the quotient rule, you divide \(2z\) by \(x\).
Thus, this expression can be rewritten as \(\log_{3} \left(\frac{2z}{x}\right)\), changing two separate logarithms into one. This operation not only simplifies the expression but also makes it more digestible.
Power Rule of Logarithms
The power rule of logarithms is another neat trick that simplifies expressions involving logs. It allows us to take a coefficient in front of a log and convert it into an exponent inside the log. This conversion is achieved through:
  • \(c \log_{b} a = \log_{b} (a^{c})\)
This means that any constant in front of the log can become an exponent on the inside part of the logarithm.
Take the expression \(\frac{1}{5} \log_{3} y\) as an example. By applying the power rule, the \(\frac{1}{5}\) becomes the exponent in \(y\), written as \(y^{1/5}\).
So, \(\frac{1}{5} \log_{3} y\) is equivalent to \(\log_{3} (y^{1/5})\). This shift simplifies the expression into a singular logarithmic term, making calculations straightforward.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In 1840 , Britain experienced a bovine (cattle and oxen) epidemic called epizooty. The estimated number of new cases every 28 days is listed in the table. At the time, the London Daily made a dire prediction that the number of new cases would continue to increase indefinitely. William Farr correctly predicted when the number of new cases would peak. Of the two functions $$\begin{array}{l}f(t)=653(1.028)^{t} \\\g(t)=54,700 e^{-(t-200)^{2} / 7500}\end{array}$$ and one models the newspaper's prediction and the other models Farr's prediction, where \(t\) is in days with \(t=0\) corresponding to August \(12,1840\). $$\begin{array}{|c|c|}\hline \text { Date } & \text { New cases } \\\\\hline \text { Aug. 12 } & 506 \\\\\hline \text { Sept. 9 } & 1289 \\\\\hline \text { Oct. 7 } & 3487 \\\\\hline \text { Nov. 4 } & 9597 \\\\\hline \text { Dec. 2 } & 18,817 \\\\\hline \text { Dec. 30 } & 33,835 \\\\\hline \text { Jan. 27 } & 47,191 \\\\\hline\end{array}$$ (a) Graph each function, together with the data, in the viewing rectangle \([0,400,100]\) by \([0,60,000,10,000]\) (b) Determine which function better models Farr's prediction. (c) Determine the date on which the number of new cases peaked.

Urban population density An urban density model is a formula that relates the population density \(D\) (in thousands/mi \(^{2}\) ) to the distance \(x\) (in miles) from the center of the city. The formula \(D=a e^{-b x}\) for central density \(a\) and coefficient of decay \(b\) has been found to be appropriate for many large U.S. cities. For the city of Atlanta in 1970 , \(a=5.5\) and \(b=0.10 .\) At approximately what distance was the population density 2000 per square mile?

Exer. \(87-88:\) Graph \(f\) and \(g\) on the same coordinate plane, and estimate the solution of the inequality \(f(x) \geq g(x)\) $$f(x)=x \ln |x|: \quad g(x)=0.15 e^{x}$$

Exer. \(77-78:\) Graph \(f\) and \(g\) on the same coordinate plane, and estimate the solution of the inequality \(f(x)>g(x)\) $$f(x)=3^{-x}-4^{0.2 x} ; \quad g(x)=\ln (1.2)-x$$

Drug absorption If a 100 -milligram tablet of an asthma drug is taken orally and if none of the drug is present in the body when the tablet is first taken, the total amount \(A\) in the bloodstream after \(t\) minutes is predicted to be $$ A=100[1-(0.9)] \quad \text { for } \quad 0 \leq t \leq 10 $$ (a) Sketch the graph of the equation. (b) Determine the number of minutes needed for 50 milligrams of the drug to have entered the bloodstream.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.