Write each equation in its equivalent exponential form. log₃ 81 = 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 result \( a \), and the logarithm value \( c \) from the equation \( \log_3 81 = y \). Here, \( b = 3 \), \( a = 81 \), and \( c = y \).
3. Rewrite the logarithmic equation \( \log_3 81 = y \) in its equivalent exponential form using the definition: \( 3^y = 81 \).
4. This expresses the original logarithmic equation as an exponential equation, which is the required equivalent form.
5. You can verify this by recognizing that \( 81 \) is a power of \( 3 \), but the problem only asks for the equivalent exponential form, so no further calculation is needed.

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, log3 81 = y means 3 raised to the power y equals 81. Understanding this definition is essential to convert between logarithmic and exponential forms.

Exponential Form of a Logarithmic Equation

The exponential form of a logarithmic equation log_b a = c is b^c = a. This conversion is fundamental because it allows solving logarithmic equations by rewriting them as exponential equations, making it easier to find unknown exponents or values.

Properties of Exponents and Powers

Recognizing powers of numbers helps simplify and solve equations. For instance, knowing that 81 = 3^4 allows you to rewrite log3 81 = y as 3^y = 3^4, leading to y = 4. Familiarity with exponent rules aids in identifying equivalent expressions.