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Chapter 4: Problem 11

Exer. 11-28: Find the exact solution, using common logarithms, and a two- decimal-place approximation of each solution, when appropriate. $$2^{-x}=8$$

Short Answer

Expert verified
The exact solution is \(x = -3\). The two-decimal approximation is not needed as the solution is an integer.

Step by step solution

01

Recognize the Base-Exponential Relationship

Notice that the equation \(2^{-x} = 8\) involves an exponential expression where the base is 2. Our goal is to solve for \(x\).
02

Express 8 as a Power of 2

Since we know that 8 can be expressed as \(2^3\), rewrite the equation as \(2^{-x} = 2^3\).
03

Equate Exponents

Once the bases on both sides of the equation are the same, the exponents must be equal. Therefore, set \(-x = 3\).
04

Solve for x

Solve the equation \(-x = 3\) for \(x\) by multiplying both sides by -1, resulting in \(x = -3\).
05

Verify by Substitution

To verify the solution, substitute \(x = -3\) back into the original equation: \(2^{-(-3)} = 2^3 = 8\), confirming that our solution is correct.

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Key Concepts

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

Common Logarithms
Logarithms are a handy tool when solving exponential equations. Common logarithms refer to logarithms with the base 10. They help simplify complex exponential expressions by converting them into a more manageable form that involves addition or multiplication. When dealing with exponential equations where the base can鈥檛 be easily matched on both sides, common logarithms are often used to transform the equation into a linear one. This allows us to isolate the variable, making it easier to solve for unknown values. In our exercise, however, 2 is the base for both sides, so there was no need for common logarithms.
Base-Exponential Relationship
Understanding the base-exponential relationship is crucial in solving exponential equations. The base refers to the number being multiplied, and the exponent shows how many times the base is used as a factor.
This relationship tells us that if two exponential expressions have the same base, their exponents must be equal. In the given problem, we have the equation \(2^{-x} = 8\). By knowing that \(8\) can be rewritten as \(2^3\), we transform the equation into \(2^{-x} = 2^3\).
Now, since the bases are equal, we can equate the exponents: \(-x = 3\). It's important because it allows simplifying the equation dramatically, focusing directly on the exponents.
Equating Exponents
Once the bases in an exponential equation are identical, you can equate their exponents to find the solution. This step is sometimes referred to as setting the exponents equal. Equating exponents is a critical shortcut that helps bypass more complex algebraic manipulations.
In our case, once we've written \(8\) as \(2^3\), the equation becomes \(2^{-x} = 2^3\). With identical bases, it simplifies to \(-x = 3\). Solving \(-x = 3\) gives us \(x = -3\) by multiplying both sides with \(-1\). This approach saves time and reduces the risk of errors.
Verification by Substitution
Verification by substitution is the final step to ensure your solution is correct. After finding \(x = -3\), substitute it back into the original equation to check if it satisfies the equation.
Replace \(x\) in \(2^{-x} = 8 \) with \(-3\), and you get \(2^{-(-3)} = 2^3 = 8\). Since both sides of the original equation are equal, it confirms that \(x = -3\) is indeed the correct solution.
This step is important because it provides proof that your solution makes the equation true, minimizing errors and giving confidence in your result.

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