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Chapter 10: Problem 45

Solve each equation. $$ \left(\frac{3}{2}\right)^{x}=\frac{8}{27} $$

Short Answer

Expert verified
x = -3

Step by step solution

01

Understanding the Equation

The given equation is \ \ \[ \left( \frac{3}{2}\right)^{x}=\frac{8}{27} \]
02

Rewrite the Equation Using Powers

The fraction \ \ \( \frac{8}{27} \) can be rewritten as \ \ \( \left( \frac{2}{3} \right)^{3} \), since \( 8 = 2^3 \) and \( 27 = 3^3 \). \ \ So, the equation becomes: \[ \left( \frac{3}{2}\right)^{x}=\left( \frac{2}{3}\right)^{3} \]
03

Apply the Property of Negative Exponents

\ \ \( \frac{2}{3} \) can also be written as \ \ \( \frac{3}{2}^{-1} \). Therefore, the equation can be rewritten as: \ \ \ \[ \left( \frac{3}{2}\right)^{x}=\left( \frac{3}{2}\right)^{-3} \]
04

Equate the Exponents

Since the bases are now the same, we can equate the exponents of \( \frac{3}{2} \): \ \ \ \[ x = -3 \]

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Key Concepts

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

Negative Exponents
Negative exponents might seem confusing at first, but they are quite simple once you grasp the idea. When you see a negative exponent, like in \( a^{-n} \), it means you take the reciprocal of the base and then apply the positive exponent. So, \( a^{-n} = \frac{1}{a^n} \).

For example: \( 2^{-3} \) is the same as \( \frac{1}{2^3} = \frac{1}{8} \).

In our exercise, to solve \( \frac{8}{27} \), we rewrite it as \( \frac{3}{2}^{-3} \). This helps in matching both sides of the equation with the same base.
Equating Exponents
After rewriting the given equation \( \frac{3}{2}^x = \frac{8}{27} \) to \( \frac{3}{2}^x = \frac{3}{2}^{-3} \), we can equate the exponents.

When two expressions with the same base are equal, their exponents must also be equal. So in our case, if \( \frac{3}{2}^x = \frac{3}{2}^{-3} \), then \( x = -3 \).
This property makes solving exponential equations easier by converting them to simpler linear equations.

\ In general, for any base \( a \, \, a^m = a^n \), means \( m = n \).
Powers of Fractions
Working with fractions raised to an exponent can be simplified using a few key rules.

If you have \((\frac{a}{b})^n = \frac{a^n}{b^n} \). So, each part of the fraction is raised to the power separately.

For example: \( (\frac{2}{3})^3 = \frac{2^3}{3^3} = \frac{8}{27} \).

In the exercise, rewriting \( \frac{8}{27} \) as \( (\frac{2}{3})^3 \) helped us to match the bases with the original equation.
By converting to the power of a fraction, we can simplify the problem and recognize patterns letting us solve for unknowns easier.

This approach reaffirms that understanding the properties of exponents and fractions assist in solving more complicated equations.

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Most popular questions from this chapter

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ \ln e^{0.04 x}=\sqrt{3} $$

Solve each equation. Give exact solutions. $$ \log _{6}(4 x+2)=2 $$

Solve each equation. Give exact solutions. $$ \log _{3} x+\log _{3}(2 x+5)=1 $$

Use the change-of-base rule (with either common or natural logarithms) to approximate each logarithm to four decimal places. $$ \log _{7} 4 $$

Graph each logarithmic function. $$g(x)=\log _{1 / 4} x$$
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