In Exercises 79–84, solve for y.
79. 3^y = 2^(y+1)

Verified step by step guidance

1. Start with the given equation: \(3^{y} = 2^{y+1}\).
2. Rewrite the right side using exponent rules: \(2^{y+1} = 2^{y} \cdot 2^{1} = 2^{y} \cdot 2\).
3. Express the equation as \(3^{y} = 2 \cdot 2^{y}\).
4. Divide both sides by \$2^{y}\( to isolate terms involving \)y$: \(\frac{3^{y}}{2^{y}} = 2\).
5. Rewrite the left side as a single exponential: \(\left(\frac{3}{2}\right)^{y} = 2\). Then, take the natural logarithm of both sides to solve for \(y\): \(y \cdot \ln\left(\frac{3}{2}\right) = \ln(2)\).

Key Concepts

Exponential Equations

Exponential equations involve variables in the exponent position, such as 3^y or 2^(y+1). Solving these requires techniques to isolate the variable, often by rewriting the equation or applying logarithms.

Properties of Logarithms

Logarithms are the inverse operations of exponentials and help solve equations where the variable is an exponent. Key properties include log(a^b) = b log(a), which allows bringing down exponents to solve for the variable.

Equating and Simplifying Exponents

When bases differ and cannot be rewritten to a common base, taking logarithms on both sides allows comparison of exponents. Simplifying the resulting equation leads to isolating the variable and finding its value.