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. log￬√3 81 = 8

Verified step by step guidance

1. Identify the given form: The statement is in logarithmic form, \( \log_{\sqrt{3}} 81 = 8 \).
2. Recall the definition of a logarithm: \( \log_b a = c \) means \( b^c = a \).
3. Apply the definition to rewrite the logarithmic statement in exponential form.
4. In this case, \( b = \sqrt{3} \), \( a = 81 \), and \( c = 8 \).
5. Thus, the equivalent exponential form is \( (\sqrt{3})^8 = 81 \).

Key Concepts

Exponential Form

Exponential form expresses a number as a base raised to a power, represented as b^y = x, where b is the base, y is the exponent, and x is the result. This form is essential for understanding how logarithms work, as logarithmic statements can be converted into exponential statements to reveal the relationship between the base, exponent, and result.

Logarithmic Form

Logarithmic form is the inverse of exponential form, expressed as log_b(x) = y, meaning that b raised to the power of y equals x. This form is crucial for solving equations involving logarithms, as it allows us to find the exponent when the base and the result are known, facilitating the conversion between the two forms.

Change of Base Formula

The change of base formula allows for the conversion of logarithms from one base to another, expressed as log_b(a) = log_k(a) / log_k(b) for any positive k. This concept is important when dealing with logarithmic equations, as it enables simplification and calculation using more familiar bases, such as 10 or e, making it easier to solve logarithmic expressions.