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Chapter 4: Problem 57

Solve each exponential equation and check your answer by substituting into the original equation. $$8^{x+2}=32$$

Short Answer

Expert verified
The solution is \(x = -\frac{1}{3}\).

Step by step solution

01

Convert the Terms to the Same Base

The equation given is \(8^{x+2} = 32\). To solve this equation, we should first express both 8 and 32 as powers of 2. We know that \(8 = 2^3\) and \(32 = 2^5\). This allows us to rewrite the equation as \((2^3)^{x+2} = 2^5\).
02

Apply the Power of a Power Rule

Using the rule \((a^m)^n = a^{m \cdot n}\), we simplify \((2^3)^{x+2}\) to \(2^{3(x+2)}\). The equation now becomes \(2^{3(x+2)} = 2^5\).
03

Equate the Exponents

Since the bases are the same, we can equate the exponents: \(3(x+2) = 5\).
04

Solve for \(x\)

To solve \(3(x+2) = 5\), first expand the left side: \(3x + 6 = 5\). Then, subtract 6 from both sides to get \(3x = -1\). Divide both sides by 3 to isolate \(x\), yielding \(x = -\frac{1}{3}\).
05

Check the Solution

Substitute \(x = -\frac{1}{3}\) back into the original equation to verify the solution. The equation becomes \(8^{(-\frac{1}{3}+2)} = 32\). Simplifying the exponent gives \(8^{\frac{5}{3}} = 32\). Expressing 8 as \(2^3\), we have \((2^3)^{\frac{5}{3}} = 2^5\). This simplifies to \(2^5 = 2^5\), thus confirming that the solution is correct.

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Key Concepts

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

Power of a Power Rule
Understanding the power of a power rule is key when dealing with exponential equations. This rule states that \((a^m)^n = a^{m \cdot n}\). It helps simplify expressions where a base is raised to multiple exponents.
For instance, if you have \((2^3)^{x+2}\), the power of a power rule allows you to multiply the exponents: \(2^{3(x+2)}\).
  • Step One: Identify the base and exponents.
  • Step Two: Multiply the exponents as per the rule.
This simplification helps equate both sides when solving exponential equations, as the expression becomes easier to manage.
Base Conversion
Base conversion in exponential equations involves expressing numbers with a common base, which simplifies comparisons and calculations. For example, if given numbers are powers of different bases, try to rewrite them using the same base.
In our equation, \(8^{x+2} = 32\), we convert 8 and 32 to powers of 2:
  • \(8 = 2^3\)
  • \(32 = 2^5\)
This conversion helps in solving the equation because it allows us to focus on the exponents when the bases are identical. It鈥檚 a core technique in handling exponential equations effectively.
Solving for Variables
Solving for variables in exponential equations involves isolating the variable of interest. After applying rules and conversions, you must deal with the exponents directly.
For example, once \((2^3)^{x+2} = 2^5\) is simplified to \(2^{3(x+2)} = 2^5\), you can equate the exponents:
  • Set \(3(x+2) = 5\).
  • Expand and solve for \(x\): \(3x + 6 = 5\).
  • Simplify: \(3x = -1\).
  • Divide by 3: \(x = -\frac{1}{3}\).
Always substitute back to check the solution, ensuring it satisfies the original equation. This step confirms the accuracy of your solution.

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Most popular questions from this chapter

Solve each equation using the uniqueness property of logarithms. $$\log _{3}(x+6)-\log _{3} x=\log _{3} 5$$

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The growth of a bacteria population: \(P(t)=1000 \cdot 3^{t}\) If the initial population of a common bacterium is 1000 and the population triples every day, its population is given by the formula shown, where \(P(t)\) is the total population after \(t\) days. (a) Find the total population \(12 \mathrm{hr}, 1\) day, \(1 \frac{1}{2}\) days, and 2 days later. (b) Do the outputs show the population is tripling every \(24 \mathrm{hr}\) ( 1 day)? (c) Explain why this is an increasing function. (d) Graph the function using an appropriate scale.
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