In Exercises 79–84, solve for y.
81. 9e^(2y) = = x^2

Verified step by step guidance

1. Start with the given equation: \(9e^{2y} = x^2\).
2. Isolate the exponential term by dividing both sides of the equation by 9: \(e^{2y} = \frac{x^2}{9}\).
3. To solve for \(y\), take the natural logarithm (ln) of both sides to undo the exponential: \(\ln\left(e^{2y}\right) = \ln\left(\frac{x^2}{9}\right)\).
4. Use the logarithm property \(\ln\left(e^a\right) = a\) to simplify the left side: \(2y = \ln\left(\frac{x^2}{9}\right)\).
5. Finally, solve for \(y\) by dividing both sides by 2: \(y = \frac{1}{2} \ln\left(\frac{x^2}{9}\right)\).

Key Concepts

Exponential Functions

An exponential function involves a variable in the exponent, such as e^(2y). Understanding how to manipulate and isolate the variable in the exponent is essential for solving equations involving exponential expressions.

Logarithmic Functions and Properties

Logarithms are the inverse operations of exponentials. Applying logarithms allows us to solve for variables in the exponent by converting exponential equations into linear ones, making it easier to isolate and solve for y.

Algebraic Manipulation

Solving for y requires careful algebraic steps, including isolating terms, simplifying expressions, and applying inverse operations. Mastery of algebraic manipulation ensures accurate and efficient problem-solving.