Textbook Question
Solve each equation. Round answers to the nearest hundredth as needed. x2/3 =36
Ch. 4 - Inverse, Exponential, and Logarithmic Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 5, Problem 7
Solve each equation. Round answers to the nearest hundredth as needed. (1/4)x=64
Verified step by step guidance1
Recognize that the equation is an exponential equation of the form \(\left(\frac{1}{4}\right)^x = 64\).
Rewrite both sides of the equation with the same base if possible. Note that \(\frac{1}{4}\) can be written as \(4^{-1}\), and \(64\) can be expressed as a power of 4 since \(64 = 4^3\).
Substitute these expressions back into the equation to get \(\left(4^{-1}\right)^x = 4^3\).
Use the power of a power property: \(\left(a^m\right)^n = a^{mn}\), so rewrite the left side as \(4^{-x} = 4^3\).
Since the bases are the same and the expressions are equal, set the exponents equal to each other: \(-x = 3\). Then solve for \(x\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Equations
Exponential equations involve variables in the exponent position, such as (1/4)^x = 64. Solving these requires understanding how to manipulate and isolate the variable in the exponent to find its value.
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Solving Exponential Equations Using Logs
Properties of Exponents
Properties of exponents, like a^(m) * a^(n) = a^(m+n) and (a^m)^n = a^(mn), help simplify and rewrite expressions. Recognizing equivalent bases allows rewriting both sides of the equation to solve for the exponent.
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04:06Rational Exponents
Logarithms
Logarithms are the inverse operations of exponentials and are used to solve equations where the variable is an exponent. Applying logarithms to both sides helps isolate the exponent and solve for the variable.
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7:30Logarithms Introduction
Related Practice
Textbook Question
Fill in the blank(s) to correctly complete each sentence. If ƒ(x) = 4x, then ƒ(2) = and ƒ(-2) = ________.
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Textbook Question
Answer each of the following. Between what two consecutive integers must log2 12 lie?
Textbook Question
Determine whether each function graphed or defined is one-to-one.
Textbook Question
Fill in the blank(s) to correctly complete each sentence. The graph of ƒ(x) = -(1/3)x+4-5 is that of ƒ(x) = (1/3)x reflected across the ______ -axis, translated to the left ______ units and down _______ units.
Textbook Question
Answer each of the following. Write log3 12 in terms of natural logarithms using the change-of-base theorem.