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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 1 - FunctionsProblem 77c
Chapter 1, Problem 77c

Determine whether the following statements are true and give an explanation or counterexample.
log546=4log56\(\log\)_54^6=4\(\log\)_56

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Step 1: Start by applying the power rule of logarithms to the left side of the equation. The power rule states that \( \log_b(a^n) = n \cdot \log_b(a) \). Therefore, \( \log_5(4^6) = 6 \cdot \log_5(4) \).
Step 2: Now, rewrite the right side of the equation, which is \( 4 \cdot \log_5(6) \).
Step 3: Compare the expressions from Step 1 and Step 2. We have \( 6 \cdot \log_5(4) \) on the left and \( 4 \cdot \log_5(6) \) on the right.
Step 4: For the original statement to be true, \( 6 \cdot \log_5(4) \) must equal \( 4 \cdot \log_5(6) \).
Step 5: Consider whether there is a known relationship or property that equates these two expressions, or if a counterexample can be found by evaluating the expressions numerically.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Logarithmic Properties

Logarithmic properties are rules that govern the manipulation of logarithms. Key properties include the product rule, quotient rule, and power rule. For instance, the power rule states that \\log_b(a^n) = n \\log_b(a), which allows us to simplify expressions involving exponents. Understanding these properties is essential for evaluating and comparing logarithmic expressions.
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Change of Base Property

Change of Base Formula

The change of base formula allows us to convert logarithms from one base to another, expressed as \\log_b(a) = \\frac{\\log_k(a)}{\\log_k(b)} for any positive base k. This is particularly useful when dealing with logarithms that are not easily computable in their original base. It helps in simplifying complex logarithmic equations and verifying their equality.
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Change of Base Property

Exponential Equations

Exponential equations involve expressions where variables appear as exponents. Understanding how to manipulate these equations is crucial for solving logarithmic statements. For example, if \\log_b(a) = c, then it can be rewritten in exponential form as \\ b^c = a. This relationship is fundamental in proving or disproving logarithmic identities and statements.
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Related Practice
Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.


2=ln2e2=\(\ln\)2^{e}

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Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.


logbxlogby=logbx−logby\(\frac{\log_{b}\)x}{\(\log\)_{b}y}=\(\log\)_{b}x-\(\log\)_{b}y

Textbook Question

In each exercise, a function and an interval of its independent variable are given. The endpoints of the interval are associated with points PP and QQ on the graph of the function.

a. Sketch a graph of the function and the secant line through PP and QQ.

b. Find the slope of the secant line in part (a), and interpret your answer in terms of an average rate of change over the interval. Include units in your answer.

After tt seconds, an object dropped from rest falls a distance d=16t2d=16t^2, where dd is measured in feet and 2≤t≤52\(\leq{t}\]\leq{5}\).

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Textbook Question

Roots and powers Sketch a graph of the given pairs of functions. Be sure to draw the graphs accurately relative to each other.


y = (x)¹⸍³ and y = (x)¹⸍⁵

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Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.


2=10log1022=10^{\(\log\)_{10}2}

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


If f(x)=x2+1f\(\left\)(x\(\right\))=x^2+1 , then f−1(x)=1x2+1f^{-1}\(\left\)(x\(\right\))=\(\frac{1}{x^2+1}\).

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