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. 5(1.015)^x-1980 = 8

Verified step by step guidance

1. Start with the given equation: \$5(1.015)^{x-1980} = 8$.
2. Isolate the exponential expression by dividing both sides of the equation by 5: \((1.015)^{x-1980} = \frac{8}{5}\).
3. To solve for the exponent \(x - 1980\), take the natural logarithm (ln) of both sides: \(\ln\left((1.015)^{x-1980}\right) = \ln\left(\frac{8}{5}\right)\).
4. Use the logarithm power rule to bring down the exponent: \((x - 1980) \cdot \ln(1.015) = \ln\left(\frac{8}{5}\right)\).
5. Finally, solve for \(x\) by dividing both sides by \(\ln(1.015)\) and then adding 1980: \(x = \frac{\ln\left(\frac{8}{5}\right)}{\ln(1.015)} + 1980\).

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 often using logarithms to solve for the variable in the exponent.

Logarithms and Their Properties

Logarithms are the inverse operations of exponentials, allowing us to solve equations where the variable is an exponent. Key properties include log(a^b) = b*log(a), which helps isolate the variable when taking logarithms of both sides.

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.