Write each equation in its equivalent logarithmic form. 5⁴ = 625

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Identify the components of the exponential equation \(5^4 = 625\). Here, the base is 5, the exponent (or power) is 4, and the result is 625.

Recall the definition of logarithms: If \(a^b = c\), then the equivalent logarithmic form is \(\log_a c = b\), where \(a\) is the base, \(b\) is the exponent, and \(c\) is the result.

Apply this definition to the given equation by setting the base of the logarithm to 5, the argument to 625, and the result equal to the exponent 4.

Write the logarithmic form as \(\log_5 625 = 4\).

This expresses the original exponential equation in logarithmic form, showing the relationship between the base, the exponent, and the result.

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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 in exponential form can be rewritten as log_a(c) = b in logarithmic form, where 'a' is the base, 'b' is the exponent, and 'c' is the result.

Definition of a Logarithm

A logarithm answers the question: to what power must the base be raised to produce a given number? For example, log_5(625) = 4 means 5 raised to the 4th power equals 625. Understanding this definition is key to converting between forms.

Properties of Exponents

Properties of exponents, such as a^m * a^n = a^(m+n), help in manipulating and understanding exponential expressions. Recognizing these properties aids in verifying the correctness of conversions between exponential and logarithmic forms.

Write each equation in its equivalent logarithmic form. 54 = 625