In Exercises 1–8, write each equation in its equivalent exponential form. 5= logb 32

Verified step by step guidance

1. Understand that the logarithmic equation \(5 = \log_b 32\) is asking: 'To what power must the base \(b\) be raised to produce 32?'
2. Recall the definition of a logarithm: \(\log_b a = c\) is equivalent to \(b^c = a\).
3. Apply this definition to the given equation: \(5 = \log_b 32\) implies \(b^5 = 32\).
4. Rewrite the equation in exponential form: \(b^5 = 32\).
5. Verify that the exponential form correctly represents the original logarithmic equation.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. The logarithm log_b(a) answers the question: 'To what power must the base b be raised to obtain a?' This relationship is crucial for converting between logarithmic and exponential forms, as it allows us to express equations in a different but equivalent way.

Exponential Form

Exponential form refers to expressing a number as a power of a base. For example, the equation b^x = a indicates that b raised to the power x equals a. Understanding how to manipulate and convert equations into exponential form is essential for solving problems involving logarithms.

Change of Base Formula

The change of base formula allows us to convert logarithms from one base to another, which can simplify calculations. It states that log_b(a) = log_k(a) / log_k(b) for any positive k. This concept is useful when dealing with logarithmic equations, especially when the base is not easily manageable.