Write each equation in its equivalent exponential form. log₆ 216 = y

Verified step by step guidance

1. Recall the definition of logarithm: \( \log_b a = c \) means \( b^c = a \).
2. Identify the base \( b \), the argument \( a \), and the logarithm result \( c \) from the given equation \( \log_6 216 = y \). Here, \( b = 6 \), \( a = 216 \), and \( c = y \).
3. Rewrite the logarithmic equation \( \log_6 216 = y \) in its equivalent exponential form using the definition: \( 6^y = 216 \).
4. This expresses the original logarithmic equation as an exponential equation, which is the required form.
5. No further simplification is needed unless asked to solve for \( y \).

Key Concepts

Definition of Logarithms

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

Exponential Form of a Logarithmic Equation

Converting a logarithmic equation log_b(a) = c into exponential form involves rewriting it as b^c = a. This equivalence allows solving for unknowns and understanding the relationship between logarithms and exponents.

Properties of Exponents and Bases

Recognizing the base and the argument in logarithmic expressions helps in rewriting equations correctly. Knowing how to manipulate exponents and identify powers of numbers (e.g., 216 as 6^3) aids in simplifying and verifying the exponential form.