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. 0.8^x = 4

Verified step by step guidance

1. <insert step 1: Start by taking the logarithm of both sides of the equation to bring down the exponent. You can use either the common logarithm (log base 10) or the natural logarithm (ln).>
2. <insert step 2: Apply the logarithmic identity \( \log(a^b) = b \cdot \log(a) \) to the left side of the equation. This will allow you to express the exponent \( x \) as a coefficient.>
3. <insert step 3: Solve for \( x \) by isolating it on one side of the equation. This involves dividing both sides of the equation by \( \log(0.8) \).>
4. <insert step 4: Use a calculator to evaluate the logarithms and perform the division to find the decimal approximation of \( x \) to the nearest thousandth.>
5. <insert step 5: Verify your solution by substituting the value of \( x \) back into the original equation to ensure both sides are equal.>

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 0.8^x = 4, 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 0.8^x = 4, we can apply logarithmic properties to rewrite the equation in a more manageable form, allowing us to solve for x.

Rounding and Decimal Approximation

Rounding is the process of adjusting a number to a specified degree of accuracy, often to make it simpler or easier to work with. In this problem, irrational solutions must be expressed as decimals rounded to the nearest thousandth, which involves understanding how to round numbers correctly and the significance of each digit in the decimal representation.