Math Problem Statement

Apply the power property of logarithms. Assume that all variable expressions represent positive real numbers. log((4t - 3)^4) = ?

Solution

The problem asks to apply the power property of logarithms to the expression:

log((4t3)4)\log \left( (4t - 3)^4 \right)

The power property of logarithms states that:

logb(xn)=nlogb(x)\log_b(x^n) = n \cdot \log_b(x)

Applying this property to the expression:

log((4t3)4)=4log(4t3)\log \left( (4t - 3)^4 \right) = 4 \cdot \log(4t - 3)

Thus, the answer is:

4log(4t3)4 \cdot \log(4t - 3)

Let me know if you would like further explanation or have any questions.

Here are five related questions to consider:

  1. How can you simplify the logarithmic expression log(5x3)\log(5x^3)?
  2. What happens if you apply the power property to a logarithm with a fraction inside?
  3. How do you simplify log(anbm)\log(a^n \cdot b^m) using logarithmic properties?
  4. What is the result of applying the power property to a logarithmic expression with multiple terms?
  5. How do you combine log(a)+log(b)\log(a) + \log(b) into a single logarithmic expression?

Tip: Always check if the expression inside the logarithm is positive before simplifying, as logarithms are only defined for positive values.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Power Property of Logarithms

Formulas

log_b(x^n) = n * log_b(x)

Theorems

Power Property of Logarithms

Suitable Grade Level

Grades 9-11