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. 3(2)^x-2 + 1 = 100

Verified step by step guidance

1. Start by isolating the exponential expression on one side of the equation. Subtract 1 from both sides to get: \$3(2)^{x-2} = 100 - 1$.
2. Simplify the right side: \$3(2)^{x-2} = 99$.
3. Divide both sides by 3 to isolate the exponential term: \((2)^{x-2} = \frac{99}{3}\).
4. Simplify the fraction: \((2)^{x-2} = 33\).
5. To solve for \(x\), take the logarithm base 2 of both sides: \(x - 2 = \log_2(33)\). Then, solve for \(x\) by adding 2 to both sides: \(x = 2 + \log_2(33)\).

Key Concepts

Exponential Equations

Exponential equations involve variables in the exponent, such as 2^(x-2). Solving these requires isolating the exponential expression and often applying logarithms to find the variable's value.

Logarithms and Their Properties

Logarithms are the inverse operations of exponentials and are used to solve equations where the variable is an exponent. Understanding properties like log(a^b) = b log(a) helps in simplifying and solving these equations.

Rounding and Decimal Approximations

When solutions are irrational, they are often expressed as decimal approximations rounded to a specified place, such as the nearest thousandth. This involves using a calculator and understanding rounding rules to present the answer correctly.