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.