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. 4^(x-1) = 3^2x

Verified step by step guidance

1. Start with the given equation: \$4^{x-1} = 3^{2x}$.
2. Rewrite the bases as powers of prime factors if possible. Note that \$4\( can be written as \)2^2\(, so rewrite the equation as: \)\left(2^2\right)^{x-1} = 3^{2x}$.
3. Use the power of a power property: \(\left(a^m\right)^n = a^{mn}\), to simplify the left side: \$2^{2(x-1)} = 3^{2x}$.
4. Take the natural logarithm (or log base 10) of both sides to bring down the exponents: \(\ln\left(2^{2(x-1)}\right) = \ln\left(3^{2x}\right)\).
5. Apply the logarithm power rule: \(\ln(a^b) = b \ln(a)\), to get: \$2(x-1) \ln(2) = 2x \ln(3)\(. Then solve this linear equation for \)x\( by expanding and isolating \)x$.

Key Concepts

Exponential Equations

Exponential equations involve variables in the exponent position, such as 4^(x-1) = 3^(2x). Solving these requires understanding how to manipulate and equate expressions with different bases, often by applying logarithms or rewriting terms with common bases.

Logarithms and Their Properties

Logarithms are the inverse operations of exponentials and are essential for solving equations where the variable is an exponent. Using properties like log(a^b) = b*log(a) allows us to bring down exponents and solve for the variable algebraically.

Rounding and Exact vs. Approximate Solutions

Some solutions to equations are irrational and cannot be expressed exactly as fractions or decimals. Understanding when to provide exact forms (like logarithmic expressions) versus decimal approximations rounded to a specific place value is important for clear and accurate answers.