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Chapter 4: Problem 96

Solve each equation. Find the exact solutions. $$\frac{1}{2^{x}}=5$$

Short Answer

Expert verified
x = -\frac{\text{ln}(5)}{\text{ln}(2)}

Step by step solution

01

Rewrite the Equation in Exponential Form

Given the equation \(\frac{1}{2^x} = 5\), it can be rewritten as \(2^{-x} = 5\). This is done by bringing the exponent to the numerator.
02

Take the Natural Logarithm of Both Sides

Next, take the natural logarithm (ln) of both sides to simplify the equation: \(\text{ln}(2^{-x}) = \text{ln}(5)\).
03

Use the Power Rule of Logarithms

Apply the power rule of logarithms which states \(\text{ln}(a^b) = b \text{ln}(a)\). Therefore, the equation becomes \(-x \text{ln}(2) = \text{ln}(5)\).
04

Solve for x

Finally, solve for \(x\) by dividing both sides by \(-\text{ln}(2)\): \(-x = \frac{\text{ln}(5)}{\text{ln}(2)}}\), hence \(x = -\frac{\text{ln}(5)}{\text{ln}(2)}\).

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Key Concepts

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

Exponential Equations
Exponential equations involve variables in the exponent. For example, in the equation \(\frac{1}{2^x} = 5\), the variable \(x\) is in the exponent of 2. To solve this, we often need to rewrite or manipulate the equation so we can handle the exponent more easily. In our example, we rewrite \(\frac{1}{2^x}\) as \(2^{-x}\), making the exponent more apparent. Recognizing and rewriting exponential equations correctly is the first key step towards solving them.
Natural Logarithm
The natural logarithm, denoted as \(\text{ln}\), is helpful when solving exponential equations. It is the logarithm to the base \(e\), where \(e\) is approximately 2.71828. Taking the natural logarithm of both sides of an equation can help transform an exponential equation into a linear one. For example, taking \(\text{ln}(2^{-x}) = \text{ln}(5)\) simplifies the exponentiation on the left side. The natural logarithm is powerful because it allows us to bring exponents down using properties of logarithms.
Power Rule of Logarithms
The power rule of logarithms states that \( \text{ln}(a^b) = b \text{ln}(a) \). This is incredibly useful when dealing with exponential equations. By applying this rule, we can transform the exponent into a coefficient, which simplifies the equation significantly. For instance, \(\text{ln}(2^{-x}) = \text{ln}(5)\) becomes \(-x \text{ln}(2) = \text{ln}(5)\). This process turns an exponent problem into a straightforward algebraic one, allowing us to solve for the variable.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying equations to find the solution. After applying the natural logarithm and the power rule, we get \(-x \text{ln}(2) = \text{ln}(5)\). To isolate \(x\), we divide both sides by \(-\text{ln}(2)\). This gives us \(x = -\frac{\text{ln}(5)}{\text{ln}(2)}\). Remember, algebraic manipulation can involve several techniques, including addition, subtraction, multiplication, and division, to get the desired form of the equation. The key is to perform operations that help isolate the variable you need to solve for.

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Most popular questions from this chapter

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