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

검증된 단계별 안내

Start with the given equation: \(e^{\sqrt{t}} = x^2\).

Take the natural logarithm (ln) of both sides to eliminate the exponential: \(\ln\left(e^{\sqrt{t}}\right) = \ln\left(x^2\right)\).

Use the logarithm property \(\ln\left(e^a\right) = a\) to simplify the left side: \(\sqrt{t} = \ln\left(x^2\right)\).

Apply the logarithm power rule on the right side: \(\sqrt{t} = 2 \ln(x)\).

Square both sides to solve for \(t\): \(t = \left(2 \ln(x)\right)^2\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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