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

Verified step by step guidance

1. 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.
2. Rewrite the given function \(y = \log_4 x + \log_4 x^2\) as \(y = \frac{\ln x}{\ln 4} + \frac{\ln x^2}{\ln 4}\).
3. Use the logarithm power rule on \(\ln x^2\) to simplify it to \(2 \ln x\), so the function becomes \(y = \frac{\ln x}{\ln 4} + \frac{2 \ln x}{\ln 4}\).
4. Combine the terms to get \(y = \frac{\ln x + 2 \ln x}{\ln 4} = \frac{3 \ln x}{\ln 4}\).
5. 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, \(\frac{dy}{dx} = \frac{3}{\ln 4} \cdot \frac{1}{x}\).

Key Concepts

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.