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. 6^(x+1) = 4^(2x-1)

Verified step by step guidance

1. Start by rewriting the equation \$6^{x+1} = 4^{2x-1}$ and recognize that the bases 6 and 4 are different and cannot be easily rewritten as powers of the same base.
2. Take the natural logarithm (or log base 10) of both sides to use the logarithm power rule, which allows you to bring the exponents down: \(\ln(6^{x+1}) = \ln(4^{2x-1})\).
3. Apply the logarithm power rule: \((x+1) \ln(6) = (2x - 1) \ln(4)\).
4. Distribute the logarithms: \(x \ln(6) + \ln(6) = 2x \ln(4) - \ln(4)\).
5. Collect all terms involving \(x\) on one side and constants on the other side, then factor out \(x\): \(x \ln(6) - 2x \ln(4) = - \ln(4) - \ln(6)\), which simplifies to \(x (\ln(6) - 2 \ln(4)) = - (\ln(4) + \ln(6))\). Finally, solve for \(x\) by dividing both sides by \((\ln(6) - 2 \ln(4))\).

Key Concepts

Exponential Equations

Exponential equations involve variables in the exponent position, such as 6^(x+1) = 4^(2x-1). Solving these requires understanding how to manipulate and equate expressions with different bases or exponents.

Logarithms and Their Properties

Logarithms are the inverse operations of exponentials and are used to solve equations where the variable is an exponent. Applying logarithms allows us to rewrite the equation in a form that isolates the variable for easier solving.

Approximating Irrational Solutions

Some exponential equations yield irrational solutions that cannot be expressed exactly as fractions. These solutions are approximated as decimals, often rounded to a specified place value, such as the nearest thousandth.