Math Problem Statement

Write the exponential equation in logarithmic form.

8

2

=

1

64

log

8

=

2

Solution

The given exponential equation is:

82=1648^{-2} = \frac{1}{64}

To rewrite this in logarithmic form, we can use the relationship between logarithms and exponents. The general form is:

ab=cis equivalent tologa(c)=ba^b = c \quad \text{is equivalent to} \quad \log_a(c) = b

In this case, the base a=8a = 8, the exponent b=2b = -2, and the result c=164c = \frac{1}{64}.

So, the logarithmic form of the given equation is:

log8(164)=2\log_8\left(\frac{1}{64}\right) = -2


Would you like further details or explanations? Let me know!

Here are 5 related questions for you:

  1. What is the relationship between exponential and logarithmic functions?
  2. How do you solve logarithmic equations?
  3. What are the properties of logarithms?
  4. How can you express a fraction like 164\frac{1}{64} as a power of 8?
  5. What does logb(c)=x\log_b(c) = x mean in terms of exponents?

Tip: When converting from exponential to logarithmic form, remember that the base of the exponential equation becomes the base of the logarithm!

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Math Problem Analysis

Mathematical Concepts

Exponents
Logarithms
Algebra

Formulas

a^b = c ⇔ log_a(c) = b

Theorems

Logarithmic Definition Theorem

Suitable Grade Level

Grades 8-10