In Exercises 59–86, find the derivative of y with respect to the given independent variable.
85. y = log₂(8t^(ln 2))

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Recall the change of base formula for logarithms: \(\log_a b = \frac{\ln b}{\ln a}\). Use this to rewrite \(y = \log_2 \left(8t^{\ln 2}\right)\) as \(y = \frac{\ln \left(8t^{\ln 2}\right)}{\ln 2}\).

Apply the logarithm property \(\ln(ab) = \ln a + \ln b\) to expand the numerator: \(\ln \left(8t^{\ln 2}\right) = \ln 8 + \ln \left(t^{\ln 2}\right)\).

Use the power rule for logarithms: \(\ln \left(t^{\ln 2}\right) = (\ln 2) \cdot \ln t\). So the expression becomes \(y = \frac{\ln 8 + (\ln 2) \cdot \ln t}{\ln 2}\).

Since \(\ln 8\) and \(\ln 2\) are constants, separate the terms: \(y = \frac{\ln 8}{\ln 2} + \frac{(\ln 2) \cdot \ln t}{\ln 2}\). Simplify the second term by canceling \(\ln 2\).

Now differentiate \(y\) with respect to \(t\). The first term is constant, so its derivative is zero. For the second term, differentiate \(\ln t\) using \(\frac{d}{dt} \ln t = \frac{1}{t}\). Combine these results to find \(\frac{dy}{dt}\).

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주요 개념

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Logarithmic Functions and Their Properties

Logarithmic functions are the inverses of exponential functions. Key properties include the change of base formula and the laws of logarithms, such as log_b(xy) = log_b(x) + log_b(y). Understanding these properties helps simplify expressions before differentiation.

Chain Rule

The chain rule is used to differentiate composite functions. It states that the derivative of a function composed with another function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

Derivative of Logarithmic Functions with Arbitrary Bases

The derivative of log base b of x is 1/(x ln b). This formula generalizes the natural logarithm derivative and is essential when differentiating logarithms with bases other than e.

In Exercises 59–86, find the derivative of y with respect to the given independent variable.85. y = log₂(8t^(ln 2))