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. ${\log }_{\sqrt{3}}81=8$

Verified step by step guidance

1. Identify the given statement: \( \log_{\sqrt{3}} 81 = 8 \). This is in logarithmic form, where the base is \( \sqrt{3} \), the argument is \( 81 \), and the result (or exponent) is \( 8 \).
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: rewrite \( \log_{\sqrt{3}} 81 = 8 \) as \( (\sqrt{3})^8 = 81 \).
4. Express the exponential form clearly: the base \( \sqrt{3} \) raised to the power \( 8 \) equals \( 81 \).
5. This completes the conversion from logarithmic to exponential form.

Key Concepts

Exponential and Logarithmic Forms

Exponential and logarithmic forms are two ways to express the same relationship. An exponential form is written as b^x = y, where b is the base, x is the exponent, and y is the result. The equivalent logarithmic form is log_b(y) = x, which asks the question: to what power must the base b be raised to get y?

Properties of Logarithms

Logarithms have specific properties that help in rewriting and simplifying expressions. For example, the base of the logarithm must be positive and not equal to 1, and the argument (the value inside the log) must be positive. Understanding these properties ensures correct conversion between logarithmic and exponential forms.

Radicals and Exponents

Radicals like square roots can be expressed as fractional exponents, e.g., √3 = 3^(1/2). Recognizing this helps in interpreting the base of the logarithm or the exponent in the exponential form, making it easier to rewrite the statement accurately.