Math Problem Statement

f(x) = -log_2(3x)

Solution

The given function is:

f(x)=log2(3x)f(x) = -\log_2(3x)

This is a logarithmic function with a base of 2, and the argument inside the logarithm is 3x3x. The negative sign in front of the logarithm means that the values of the function will be negated.

Key Properties:

  1. Domain: Since the argument of a logarithmic function must be greater than 0, we have: 3x>0    x>03x > 0 \implies x > 0 So, the domain of the function is (0,)(0, \infty).

  2. Range: The logarithmic function log2(3x)\log_2(3x) has a range of (,)(-\infty, \infty). However, due to the negative sign, the function f(x)f(x) will output values in (,)(-\infty, \infty).

  3. Asymptote: There is a vertical asymptote at x=0x = 0, as log2(3x)\log_2(3x) \to -\infty when x0+x \to 0^+.

  4. Behavior:

    • As x0+x \to 0^+, f(x)f(x) \to \infty because of the negative logarithm.
    • As xx \to \infty, f(x)f(x) \to -\infty.

Would you like to go into further details about how to solve or analyze this function?

Here are 5 questions related to this topic:

  1. How would you find the inverse of the function f(x)=log2(3x)f(x) = -\log_2(3x)?
  2. What is the derivative of the function f(x)f(x)?
  3. Can you graph f(x)=log2(3x)f(x) = -\log_2(3x) and explain the key features?
  4. How do you solve f(x)=0f(x) = 0 for xx?
  5. What happens to the function if you change the base of the logarithm?

Tip: Logarithmic transformations can help linearize exponential data, making it easier to analyze complex relationships.

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Math Problem Analysis

Mathematical Concepts

Logarithmic Functions
Base-2 Logarithms

Formulas

log_b(x) = y implies b^y = x

Theorems

Domain of Logarithmic Functions
Range and Asymptote of Logarithmic Functions

Suitable Grade Level

Grades 10-12