In Exercises 1–8, write each equation in its equivalent exponential form. log6 216 = y

Verified step by step guidance

1. Identify the components of the logarithmic equation: \( \log_6 216 = y \).
2. Recall the definition of a logarithm: \( \log_b a = c \) means \( b^c = a \).
3. Apply the definition to the given equation: \( \log_6 216 = y \) implies \( 6^y = 216 \).
4. Rewrite the logarithmic equation in exponential form: \( 6^y = 216 \).
5. Verify the conversion by checking if the exponential form correctly represents the original logarithmic equation.

Key Concepts

Logarithms

A logarithm is the inverse operation to exponentiation, representing the power to which a base must be raised to obtain a given number. In the expression log_b(a) = c, b is the base, a is the number, and c is the logarithm. Understanding logarithms is essential for converting between logarithmic and exponential forms.

Exponential Form

Exponential form expresses a relationship where a base is raised to a power. For example, if log_b(a) = c, it can be rewritten in exponential form as b^c = a. This transformation is crucial for solving equations involving logarithms and understanding their properties.

Base of a Logarithm

The base of a logarithm indicates the number that is raised to a power. In the equation log6(216) = y, the base is 6, meaning that 6 raised to the power of y equals 216. Recognizing the base is vital for accurately converting logarithmic expressions into their exponential counterparts.