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. See Examples 1–4. (1/2)^x = 5

Verified step by step guidance

1. Take the logarithm of both sides of the equation: \( \log((1/2)^x) = \log(5) \).
2. Use the power rule of logarithms to bring down the exponent: \( x \cdot \log(1/2) = \log(5) \).
3. Solve for \( x \) by dividing both sides by \( \log(1/2) \): \( x = \frac{\log(5)}{\log(1/2)} \).
4. Calculate \( \log(5) \) and \( \log(1/2) \) using a calculator to find their decimal values.
5. Divide the decimal value of \( \log(5) \) by the decimal value of \( \log(1/2) \) to find \( x \), rounding to the nearest thousandth.

Key Concepts

Exponential Equations

Exponential equations are equations in which variables appear as exponents. To solve these equations, one often uses logarithms, which are the inverse operations of exponentiation. For example, in the equation (1/2)^x = 5, we can take the logarithm of both sides to isolate the variable x.

Logarithms

Logarithms are mathematical functions that help solve for exponents. The logarithm of a number is the exponent to which a base must be raised to produce that number. In the context of the equation (1/2)^x = 5, we can apply logarithmic properties to rewrite the equation in a more manageable form, allowing us to solve for x.

Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction, meaning their decimal representation is non-repeating and non-terminating. When solving equations like (1/2)^x = 5, the solutions may be irrational, and it is often required to express these solutions as decimals rounded to a specific precision, such as the nearest thousandth.