In Exercises 9–20, write each equation in its equivalent logarithmic form. 5^4 = 625

Verified step by step guidance

1. Identify the components of the exponential equation: the base is 5, the exponent is 4, and the result is 625.
2. Recall the definition of a logarithm: if \( b^y = x \), then \( \log_b(x) = y \).
3. Apply the definition to the given equation \( 5^4 = 625 \).
4. Rewrite the equation in logarithmic form: \( \log_5(625) = 4 \).
5. Verify that the logarithmic form correctly represents the original exponential equation.

Key Concepts

Exponential Form

Exponential form expresses a number as a base raised to a power. In the equation 5^4 = 625, 5 is the base, 4 is the exponent, and 625 is the result of the exponentiation. Understanding this form is crucial for converting to logarithmic form.

Logarithmic Form

Logarithmic form is the inverse of exponential form and expresses the relationship between the base, exponent, and result. The logarithmic form of the equation 5^4 = 625 is log base 5 of 625 equals 4, written as log₅(625) = 4. This transformation is essential for solving problems involving exponents.

Change of Base Formula

The change of base formula allows the conversion of logarithms from one base to another, which is useful when calculating logarithms on calculators that may not support all bases. The formula is logₐ(b) = logₓ(b) / logₓ(a), where x is any positive number. This concept is important for understanding how to manipulate logarithmic expressions.