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. 0.8^x = 4

Verified step by step guidance

1. Identify the equation to solve: \(0.8^x = 4\).
2. Since the variable \(x\) is in the exponent, apply the natural logarithm (or log base 10) to both sides to bring the exponent down: \(\ln(0.8^x) = \ln(4)\).
3. Use the logarithm power rule to rewrite the left side: \(x \cdot \ln(0.8) = \ln(4)\).
4. Isolate \(x\) by dividing both sides by \(\ln(0.8)\): \(x = \frac{\ln(4)}{\ln(0.8)}\).
5. Calculate the right-hand side using a calculator to find the decimal approximation of \(x\) to the nearest thousandth.

Key Concepts

Exponential Equations

An exponential equation involves variables in the exponent, such as 0.8^x = 4. Solving these requires understanding how to manipulate expressions where the unknown is an exponent, often by applying logarithms to isolate the variable.

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 to both sides of an equation allows you to bring the exponent down and solve for the variable.

Decimal Approximation of Irrational Numbers

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