If f(x)=logx 1 / 9-log3 x^2(x>1) then maxf(x) is cqual to |...

Name: Problem 59, Chapter 24: Comprehensive Mathematics for JEE Advanced
Uploaded: 2021-12-02T12:49:41-08:00
Duration: 2 min 18 s
Channel: Aman Gupta
Description: If f(x)=logx 1 / 9-log3 x^2(x>1) then maxf(x) is cqual to

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions and have distinct properties such as a restricted domain and a continuous, monotonically increasing or decreasing behavior depending on the base. They are foundational in understanding how exponential growth or decay can be transformed into linear relationships, making them useful in both pure and applied mathematics.

Change of Base Formula

The change of base formula allows a logarithm with one base to be expressed in terms of logarithms with another base. This is especially useful for simplifying expressions involving logarithms with variable bases or when different logarithmic terms need to be combined into a uniform format, which is crucial when solving equations or optimizing functions that involve logarithms.

Properties of Logarithms

Key properties such as the power rule, product rule, and quotient rule enable simplification and manipulation of logarithmic expressions. For example, the power rule states that log_b (x^n) equals n * log_b x. These properties are essential when rewriting and simplifying complex logarithmic expressions to facilitate further analysis or optimization.

Function Optimization

Function optimization involves finding the maximum or minimum values of a function, often using calculus methods like taking derivatives and setting them equal to zero to determine critical points. In the context of functions involving logarithms, transforming the expression into a form that reveals a standard optimization problem is a common strategy to identify such extrema.

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