Write each equation in its equivalent logarithmic form. ∛8 = 2

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Identify the given exponential equation: \(\sqrt[3]{8} = 2\). This can be rewritten as \(8^{\frac{1}{3}} = 2\) because the cube root of 8 is the same as raising 8 to the power of \(\frac{1}{3}\).

Recall the relationship between exponential and logarithmic forms: if \(a^x = b\), then the equivalent logarithmic form is \(\log_{a} b = x\).

In the equation \(8^{\frac{1}{3}} = 2\), identify the base \(a = 8\), the exponent \(x = \frac{1}{3}\), and the result \(b = 2\).

Apply the logarithmic form using the identified values: write \(\log_{8} 2 = \frac{1}{3}\).

This expresses the original equation in its equivalent logarithmic form, completing the conversion.

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Exponential and Logarithmic Forms

Exponential and logarithmic forms are two ways to express the same relationship. An equation like a^b = c can be rewritten as log_a(c) = b, where the logarithm answers the question: to what power must the base a be raised to get c?

Cube Roots and Rational Exponents

A cube root, such as ∛8, can be expressed as an exponent of 1/3, so ∛8 = 8^(1/3). Understanding this helps convert root expressions into exponential form, which is essential for rewriting equations in logarithmic form.

Properties of Logarithms

Logarithms have properties that allow simplification and conversion between forms. Recognizing that log_b(c) = x means b^x = c is key to rewriting equations, especially when dealing with roots and powers.