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Chapter 5: Problem 7

Solve each equation. $$\left(\frac{1}{2}\right)^{x}=4$$

Short Answer

Expert verified
The solution is \( x = -2 \).

Step by step solution

01

Express Both Sides with the Same Base

First, recognize that the equation \( \left( \frac{1}{2} \right)^x = 4 \) can be expressed with the same base. Note that \( 4 \) can be rewritten as \( 2^2 \). Then the equation becomes \[ \left( \frac{1}{2} \right)^x = 2^2. \] Also notice \( \left( \frac{1}{2} \right)^x \) can be rewritten as \( 2^{-x} \).
02

Set the Exponents Equal to Each Other

Since the bases are now the same (\( 2^{-x} = 2^2 \)), we can set the exponents equal to each other. This gives us the equation: \[ -x = 2. \]
03

Solve for x

Solve the equation \( -x = 2 \) for \( x \) by multiplying both sides by \(-1\): \[ x = -2. \]

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Key Concepts

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

Base Conversion in Exponential Equations
Base conversion is an important concept when dealing with exponential equations, especially when trying to solve them. In the exercise, we started with the equation \( \left(\frac{1}{2}\right)^x = 4 \). To work with the equation more easily, both sides were rewritten to have the same base.
  • The fraction \( \frac{1}{2} \) has the base 2, but expressed negatively as it is in the form \( 2^{-1} \).
  • The number 4 can also be expressed as \( 2^2 \).
  • By converting both sides to the base of 2, the equation becomes a comparison of the exponents: \( 2^{-x} = 2^2 \).
This transformation makes it easier to solve the equation because we only focus on the exponents once the bases are equal. Using the same base simplifies the process of comparison and elimination of exponentials.
Understanding Exponent Rules
Exponent rules are essential for solving many mathematical problems involving powers. In the previous step, we converted \( \left( \frac{1}{2} \right)^x \) to \( 2^{-x} \) using the following rules:
  • Negative Exponent Rule: Any expression \( a^{-b} \) is equivalent to \( \frac{1}{a^b} \). This converted \( \frac{1}{2} \) to \( 2^{-1} \).
  • Power of a Power Rule: When a power is raised to another power, the exponents are multiplied. For example, \( (2^{-1})^x = 2^{-x} \).
Understanding these rules allows for the conversion of more complex exponential forms into easily comparable expressions. Once you rewrite the equation using these rules, you set the exponents equal to each other when the bases match, since \( a^m = a^n \) implies \( m = n \).
Solving Exponential Equations
Once the exponential equation is simplified to a form where the bases are identical, solving it involves setting the exponents equal to each other. Starting from the equation \( 2^{-x} = 2^2 \), we can easily deduce:
  • The exponent on the left side is \( -x \).
  • The exponent on the right side is \( 2 \).
  • Therefore, \( -x = 2 \).
To solve \( -x = 2 \), perform basic algebraic steps:
  • Multiply both sides by \(-1\) to isolate \(x\). This yields \( x = -2 \).
The solution \( x = -2 \) represents the value for \( x \) that satisfies the original equation, considering all transformations and manipulations carried out. These steps reflect a core process in tackling exponential equations effectively.

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Most popular questions from this chapter

The following table shows the revenue in billions of dollars for iTunes in various years. $$\begin{array}{|l|c|c|c|c|c|} \hline \text { Year } & 2008 & 2009 & 2010 & 2011 & 2012 \\ \hline \begin{array}{l} \text { Revenue } \\ \text { (\$billions) } \end{array} & 6.0 & 7.0 & 9.5 & 11.8 & 15.7 \\\\\hline\end{array}$$ (a) Use exponential regression to approximate constants \(C\) and \(a\) so that \(f(x)=C a^{x-2008}\) models the data, where \(x\) is the year. (b) Support your answer by graphing \(f\) and the data.

Use any method (analytic or graphical) to solve each equation. $$\log _{2} \sqrt{2 x^{2}}-1=0.5$$

In \(2012,17 \%\) of the U.S. population was Hispanic, and this number is expected to be \(31 \%\) in \(2060 .\) (Source: U.S. Census Bureau.) (a) Approximate \(C\) and \(a\) so that \(P(x)=C a^{x-2012}\) models these data, where \(P\) is the percent of the population that is Hispanic and \(x\) is the year. Why is \(a>1 ?\) (b) Estimate \(P\) in 2030 . (c) Use \(P\) to estimate the year when \(25 \%\) of the population could be Hispanic.

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Use the change-of-base rule to find an approximation for each logarithm. $$\log _{15} 5$$
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