In Exercises 1–4, solve for t.
e^(sqrt(t)) = x^2

Verified step by step guidance

1. Start with the given equation: \(e^{\sqrt{t}} = x^2\).
2. Take the natural logarithm (ln) of both sides to eliminate the exponential: \(\ln\left(e^{\sqrt{t}}\right) = \ln\left(x^2\right)\).
3. Use the logarithm property \(\ln\left(e^a\right) = a\) to simplify the left side: \(\sqrt{t} = \ln\left(x^2\right)\).
4. Apply the logarithm power rule on the right side: \(\sqrt{t} = 2 \ln(x)\).
5. Square both sides to solve for \(t\): \(t = \left(2 \ln(x)\right)^2\).

Key Concepts

Exponential Functions

An exponential function involves a constant base raised to a variable exponent, such as e^(sqrt(t)). Understanding how to manipulate and invert these functions is essential for solving equations where the variable is in the exponent.

Square Root and Its Properties

The square root function, denoted sqrt(t), is the inverse of squaring a number. Recognizing how to isolate and handle the square root in an equation is crucial for simplifying and solving for the variable t.

Logarithmic Functions and Their Inverses

Logarithms are the inverse operations of exponentials. Applying the natural logarithm (ln) to both sides of an equation like e^(sqrt(t)) = x^2 allows us to 'undo' the exponential and solve for sqrt(t), facilitating the isolation of t.