In Exercises 13–15, write each equation in its equivalent exponential form. 1/2 = log49 7

Verified step by step guidance

1. Understand that the equation \( \frac{1}{2} = \log_{49} 7 \) is in logarithmic form, where \( \log_b a = c \) means \( b^c = a \).
2. Identify the base \( b \), the result \( a \), and the exponent \( c \) from the logarithmic equation. Here, \( b = 49 \), \( a = 7 \), and \( c = \frac{1}{2} \).
3. Rewrite the equation in exponential form using the identified components: \( 49^{\frac{1}{2}} = 7 \).
4. Recognize that rewriting in exponential form involves expressing the base raised to the power of the exponent equals the result.
5. Verify the conversion by checking if the exponential form correctly represents the original logarithmic equation.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. They express the power to which a base must be raised to obtain a certain number. For example, in the equation log_b(a) = c, b is the base, a is the result, and c is the exponent. Understanding this relationship is crucial for converting between logarithmic and exponential forms.

Exponential Form

Exponential form refers to expressing a number as a base raised to a power. For instance, the equation a = b^c indicates that b is raised to the power of c to yield a. In the context of logarithms, converting a logarithmic equation to exponential form allows us to solve for unknown variables by rewriting the equation in a more straightforward manner.

Change of Base Formula

The change of base formula is a method used to convert logarithms from one base to another, which can simplify calculations. It states that log_b(a) can be expressed as log_k(a) / log_k(b) for any positive k. This concept is particularly useful when dealing with logarithms that do not have a readily available base, allowing for easier computation and understanding of logarithmic relationships.