In Exercises 59–86, find the derivative of y with respect to the given independent variable.
73. y = log₄ x + log₄ x²

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Recall that the logarithm with base 4 can be rewritten using the change of base formula: \(\log_4 x = \frac{\ln x}{\ln 4}\), where \(\ln\) is the natural logarithm.

Rewrite the given function \(y = \log_4 x + \log_4 x^2\) as \(y = \frac{\ln x}{\ln 4} + \frac{\ln x^2}{\ln 4}\).

Use the logarithm power rule on \(\ln x^2\) to simplify it to \(2 \ln x\), so the function becomes \(y = \frac{\ln x}{\ln 4} + \frac{2 \ln x}{\ln 4}\).

Combine the terms to get \(y = \frac{\ln x + 2 \ln x}{\ln 4} = \frac{3 \ln x}{\ln 4}\).

Differentiate \(y\) with respect to \(x\) using the derivative of \(\ln x\), which is \(\frac{1}{x}\), and treat \(\frac{3}{\ln 4}\) as a constant multiplier. So, \(\frac{dy}{dx} = \frac{3}{\ln 4} \cdot \frac{1}{x}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Logarithmic Functions and Their Properties

Logarithmic functions are the inverses of exponential functions and have specific properties such as log_b(xy) = log_b(x) + log_b(y) and log_b(x^n) = n log_b(x). Understanding these properties helps simplify expressions before differentiation.

Change of Base Formula

The change of base formula, log_b(x) = ln(x) / ln(b), allows rewriting logarithms with any base in terms of natural logarithms, which are easier to differentiate using standard rules.

Derivative of Logarithmic Functions

The derivative of ln(x) with respect to x is 1/x. Using the chain rule and the change of base formula, the derivative of log_b(x) is 1 / (x ln(b)), which is essential for finding the derivative of logarithms with bases other than e.

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