Math Problem Statement

Type into log form

(1/4)^(-3) = 64

Solution

We are given the following equation:

(14)3=64\left( \frac{1}{4} \right)^{-3} = 64

To express this equation in logarithmic form, we first simplify the expression:

(14)3=(41)3=43=64\left( \frac{1}{4} \right)^{-3} = \left( \frac{4}{1} \right)^3 = 4^3 = 64

Now, rewriting this as a logarithmic equation, we use the logarithmic identity:

ab=ccan be written asloga(c)=ba^b = c \quad \text{can be written as} \quad \log_a(c) = b

In this case, 43=644^3 = 64, so we express it as:

log4(64)=3\log_4(64) = 3

This is the logarithmic form of the given equation.

Do you need further clarification or details? Here are some related questions for further exploration:

  1. How would you express (19)2=81(\frac{1}{9})^{-2} = 81 in logarithmic form?
  2. Can you convert 64=2664 = 2^6 into its logarithmic form?
  3. What happens when you have a negative exponent with a base less than 1?
  4. How does changing the base affect the logarithmic equation?
  5. How do you solve logarithmic equations like log4(64)=3\log_4(64) = 3?

Tip: When working with logarithms, always remember the base in both exponential and logarithmic forms, as they must remain consistent.

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Math Problem Analysis

Mathematical Concepts

Exponents
Logarithms

Formulas

a^b = c can be written as log_a(c) = b
(1/a)^(-n) = a^n

Theorems

Laws of Exponents

Suitable Grade Level

Grades 8-10