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. 0.05(1.15)^x = 5

Verified step by step guidance

1. Start with the given equation: \$0.05(1.15)^x = 5$.
2. Isolate the exponential expression by dividing both sides of the equation by 0.05: \((1.15)^x = \frac{5}{0.05}\).
3. Simplify the right side to get a numerical value: \((1.15)^x = 100\).
4. To solve for \(x\), take the logarithm of both sides. You can use the natural logarithm (ln) or common logarithm (log): \(\ln((1.15)^x) = \ln(100)\).
5. Use the logarithm power rule to bring down the exponent: \(x \cdot \ln(1.15) = \ln(100)\). Then solve for \(x\) by dividing both sides by \(\ln(1.15)\): \(x = \frac{\ln(100)}{\ln(1.15)}\).

Key Concepts

Exponential Equations

An exponential equation involves variables in the exponent, such as a^x = b. Solving these requires isolating the exponential expression and using logarithms to solve for the variable exponent.

Logarithms and Their Properties

Logarithms are the inverse operations of exponentials, allowing us to solve for exponents. Applying log to both sides of an equation helps isolate the variable in the exponent, using properties like log(a^x) = x log(a).

Rounding Decimal Answers

When solutions are irrational, they are often expressed as decimals rounded to a specified place value. Here, answers must be rounded to the nearest thousandth, meaning three digits after the decimal point.