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Chapter 12: Problem 46

$$ \left(\frac{4}{3}\right)^{x}=\frac{27}{64} $$

Short Answer

Expert verified
x = -3

Step by step solution

01

- Understand the Equation

The equation to solve is \ \( \left(\frac{4}{3}\right)^{x} = \frac{27}{64} \). We need to find the value of \(x\) that makes this equation true.
02

- Rewrite the Right-Hand Side

Rewrite \( \frac{27}{64} \) as a power of fractions: \( \frac{27}{64} = \left(\frac{3}{4}\right)^{3} \). This step is crucial for equating the bases.
03

- Flip the Base

Notice that \( \left(\frac{3}{4}\right) \) is the reciprocal of \( \frac{4}{3} \). So, \( \left(\frac{3}{4}\right) = \left(\frac{4}{3}\right)^{-1} \). This allows us to rewrite the equation again.
04

- Apply the Reciprocal

Re-write \( \left(\frac{4}{3}\right)^{-1} \) to get \( \left(\frac{4}{3}\right)^{x} = \left(\frac{4}{3}\right)^{-3} \). Now both sides have the same base.
05

- Equate Exponents

Since the bases are the same, we can equate the exponents: \( x = -3 \).

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Key Concepts

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

exponents
Exponents are a way to express repeated multiplication of a number by itself. When we say \(a^n\), we mean that \a\ is multiplied by itself \ times. For example, \2^3\ means \2\ * \2\ * \2\. This is very helpful in simplifying large numbers and performing operations efficiently. In our example, the equation \(\left(\frac{4}{3}\right)^x = \frac{27}{64}\) involves exponents. Understanding how to manipulate and solve equations involving exponents is crucial.
fractions
Fractions represent parts of a whole. They consist of a numerator and a denominator. In the fraction \(\frac{a}{b}\), \a\ is the numerator and \b\ is the denominator. Fractions can be rewritten as exponents to simplify equations. In our problem, \(\frac{27}{64}\) is a fraction that we converted to an exponent form \(\left(\frac{3}{4}\right)^3\). Identifying and converting fractions into exponential forms can make solving equations easier.
reciprocal
The reciprocal of a number is one divided by that number. For a fraction \(\frac{a}{b}\), the reciprocal is \(\frac{b}{a}\). This concept is useful when solving equations involving exponents. In our problem, we recognized that \(\left(\frac{3}{4}\right)\) is the reciprocal of \(\left(\frac{4}{3}\right)\). Thus, \(\left(\frac{3}{4}\right) = \left(\frac{4}{3}\right)^{-1}\). Using reciprocals helps us match the bases on both sides of the equation before equating the exponents.

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Most popular questions from this chapter

Use the properties of logarithms to express each logarithm as a sum or difference of logarithms, or as a single logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{5} 7^{4}$$

To four decimal places, the values of \(\log _{10} 2\) and \(\log _{10} 9\) are $$\log _{10} 2=0.3010 \text { and } \log _{10} 9=0.9542$$ Use these values and the properties of logarithms to evaluate each expression. DO NOT USE A CALCULATOR. $$\log _{10} \frac{9}{2}$$

Use the properties of logarithms to write each expression as a single logarithm. Assume that all variables are defined in such a way that the variable expressions are positive, and bases are positive numbers not equal to 1. $$\log _{10}(y+4)+\log _{10}(y-4)$$

Solve each equation. \(\log _{x} 9=\frac{1}{2}\)

Use the change-of-base rule (with either common or natural logarithms) to approximate each logarithm to four decimal places. \(\log _{\pi} e\)
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