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 1–4.5(1.015)^(x-1980) = 8

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Step 1: Start by isolating the exponential expression. Divide both sides of the equation by 5 to get \((1.015)^{x-1980} = \frac{8}{5}\).

Step 2: To solve for \(x\), take the natural logarithm of both sides: \(\ln((1.015)^{x-1980}) = \ln(\frac{8}{5})\).

Step 3: Use the logarithmic identity \(\ln(a^b) = b \cdot \ln(a)\) to bring down the exponent: \((x-1980) \cdot \ln(1.015) = \ln(\frac{8}{5})\).

Step 4: Solve for \(x-1980\) by dividing both sides by \(\ln(1.015)\): \(x-1980 = \frac{\ln(\frac{8}{5})}{\ln(1.015)}\).

Step 5: Finally, solve for \(x\) by adding 1980 to both sides: \(x = 1980 + \frac{\ln(\frac{8}{5})}{\ln(1.015)}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponential Equations

Exponential equations involve variables in the exponent, such as the equation given. To solve these, one typically isolates the exponential expression and may use logarithms to bring the variable down from the exponent. Understanding the properties of exponents and logarithms is crucial for manipulating and solving these types of equations.

Logarithms

Logarithms are the inverse operations of exponentiation. They allow us to solve for the exponent in an exponential equation. For example, if we have an equation of the form a^b = c, we can use logarithms to express b as log_a(c). Familiarity with the properties of logarithms, such as the product, quotient, and power rules, is essential for solving exponential equations.

Rounding and Decimal Approximation

When dealing with irrational solutions, it is often necessary to provide decimal approximations. Rounding to a specified number of decimal places, such as the nearest thousandth, requires understanding how to round numbers correctly. This skill is important for presenting solutions in a clear and standardized format, especially in contexts where exact values are impractical.

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