Math Problem Statement

Log4 9> log4 x.Find x

Solution

To solve the equation log49>log4x\log_4{9} > \log_4{x}, follow these steps:

Step 1: Apply the property of logarithms

The inequality involves logarithms with the same base, so you can drop the logs and directly compare the arguments. This property can be applied because the logarithmic function is increasing for bases greater than 1 (here, base 4).

log49>log4x9>x\log_4{9} > \log_4{x} \quad \Rightarrow \quad 9 > x

Step 2: Write the solution

The inequality simplifies to: x<9x < 9

Thus, the solution is x(0,9)x \in (0, 9), where xx must be greater than 0 because the logarithm is not defined for non-positive numbers.

Would you like further clarification or more details on any step?

Here are 5 related questions to expand your understanding:

  1. How do you solve logarithmic inequalities with different bases?
  2. What are the properties of logarithmic functions?
  3. How can you rewrite a logarithmic inequality in exponential form?
  4. What happens to inequalities when logarithmic functions are decreasing?
  5. How do logarithms relate to exponents?

Tip: Always remember to check the domain of the logarithmic function when solving equations or inequalities involving logarithms.

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Math Problem Analysis

Mathematical Concepts

Logarithmic Inequalities
Logarithmic Functions

Formulas

log_b(a) > log_b(c) implies a > c for b > 1

Theorems

Logarithmic function properties for inequalities
Domain of logarithmic functions

Suitable Grade Level

Grades 9-12