Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. (1/2)^x = 5

Verified step by step guidance

1. Recognize that the equation is an exponential equation of the form \(\left(\frac{1}{2}\right)^x = 5\).
2. Rewrite the equation to isolate the variable \(x\) by taking the logarithm of both sides. You can use the natural logarithm (ln) or common logarithm (log):
3. \[ \ln\left(\left(\frac{1}{2}\right)^x\right) = \ln(5) \]
4. Use the logarithmic power rule to bring the exponent \(x\) in front of the logarithm:
5. \[ x \cdot \ln\left(\frac{1}{2}\right) = \ln(5) \]
6. Solve for \(x\) by dividing both sides by \(\ln\left(\frac{1}{2}\right)\):
7. \[ x = \frac{\ln(5)}{\ln\left(\frac{1}{2}\right)} \]

Key Concepts

Exponential Equations

An exponential equation involves variables in the exponent, such as (1/2)^x = 5. Solving these requires understanding how to manipulate expressions where the unknown is an exponent, often by applying logarithms or rewriting bases.

Logarithms and Their Properties

Logarithms are the inverse operations of exponentials and are used to solve equations where the variable is an exponent. Applying logarithms allows you to bring the exponent down and solve for the variable using properties like log(a^b) = b log(a).

Decimal Approximation of Irrational Numbers

When solutions involve irrational numbers, they cannot be expressed exactly as fractions. Instead, they are approximated as decimals to a specified precision, such as to the nearest thousandth, to provide a practical numerical answer.