Write each equation in its equivalent exponential form. 5= log_b 32

Verified step by step guidance

1. Recall the definition of logarithm: if \(y = \log_b x\), then the equivalent exponential form is \(b^y = x\).
2. Identify the parts of the given equation \(5 = \log_b 32\): here, \(y = 5\), \(b\) is the base, and \(x = 32\).
3. Apply the definition by rewriting the logarithmic equation \(5 = \log_b 32\) as an exponential equation: \(b^5 = 32\).
4. This expresses the original logarithmic statement in exponential form, showing the relationship between the base \(b\), the exponent \(5\), and the result \(32\).
5. No further simplification is needed unless you are asked to solve for \(b\) or another variable.

Key Concepts

Definition of Logarithms

A logarithm answers the question: to what exponent must the base be raised to produce a given number? In the expression log_b(a) = c, b is the base, a is the result, and c is the exponent such that b^c = a.

Converting Logarithmic to Exponential Form

To rewrite a logarithmic equation log_b(a) = c in exponential form, express it as b^c = a. This conversion is fundamental for solving equations involving logarithms by switching between forms.

Properties of Exponents

Understanding how exponents work, such as the meaning of b^c and how to manipulate exponential expressions, is essential when converting between logarithmic and exponential forms and solving related equations.