If the statement is in exponential form, write it in an equivalent logarithmic form. If the statement is in logarithmic form, write it in exponential form. log₄ 1/64 = -3

Verified step by step guidance

1. Identify the given logarithmic statement: \(\log_{4} \frac{1}{64} = -3\).
2. Recall the relationship between logarithmic and exponential forms: \(\log_{b} a = c\) is equivalent to \(b^{c} = a\).
3. Apply this relationship to the given statement by setting the base \(b = 4\), the result of the logarithm \(c = -3\), and the argument \(a = \frac{1}{64}\).
4. Rewrite the logarithmic equation in exponential form as \(4^{-3} = \frac{1}{64}\).
5. This expresses the original logarithmic statement in its equivalent exponential form.

Key Concepts

Definition of Logarithms

A logarithm answers the question: to what exponent must the base be raised to produce a given number? Formally, if log_b(a) = c, then b^c = a. Understanding this definition is essential to convert between logarithmic and exponential forms.

Conversion Between Exponential and Logarithmic Forms

Exponential and logarithmic forms are two ways of expressing the same relationship. The exponential form b^c = a corresponds to the logarithmic form log_b(a) = c. Recognizing this equivalence allows one to rewrite expressions from one form to the other.

Properties of Logarithms and Exponents with Fractions and Negative Exponents

Understanding how exponents work with fractions and negative numbers is crucial. For example, a negative exponent indicates a reciprocal, and fractional bases raised to powers can produce fractions. This knowledge helps interpret and verify expressions like log_4(1/64) = -3.