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. 0.05(1.15)^x = 5

Verified step by step guidance

1. <Step 1: Start by isolating the exponential expression. Divide both sides of the equation by 0.05 to get (1.15)^x = 100.>
2. <Step 2: To solve for x, take the logarithm of both sides. You can use either the natural logarithm (ln) or the common logarithm (log). For example, log((1.15)^x) = log(100).>
3. <Step 3: Apply the power rule of logarithms, which states that log(a^b) = b*log(a). This gives you x*log(1.15) = log(100).>
4. <Step 4: Solve for x by dividing both sides of the equation by log(1.15). This results in x = log(100) / log(1.15).>
5. <Step 5: Use a calculator to evaluate the expression log(100) / log(1.15) to find the value of x, rounding to the nearest thousandth as needed.>

Key Concepts

Exponential Equations

Exponential equations are equations in which variables appear as exponents. To solve these equations, one often uses logarithms to isolate the variable. In the given equation, the term (1.15)^x indicates that x is in the exponent, which requires logarithmic manipulation to solve for x.

Logarithms

Logarithms are the inverse operations of exponentiation and are used to solve equations where the variable is an exponent. The logarithm of a number is the exponent to which a base must be raised to produce that number. In this case, applying logarithms will help to transform the exponential equation into a linear form, making it easier to isolate and solve for x.

Rounding and Decimal Approximation

Rounding is the process of reducing the number of digits in a number while maintaining its value as closely as possible. In this exercise, solutions must be provided as decimals rounded to the nearest thousandth, which means keeping three digits after the decimal point. Understanding how to round correctly is essential for presenting the final answer in the required format.