In Exercises 21–42, evaluate each expression without using a calculator. 8^(log8 19)

Verified step by step guidance

1. Recognize that the expression is in the form of \( a^{\log_a b} \).
2. Recall the property of logarithms: \( a^{\log_a b} = b \).
3. Apply this property to the given expression: \( 8^{\log_8 19} \).
4. Since the base of the exponent and the base of the logarithm are the same, the expression simplifies to the argument of the logarithm.
5. Therefore, the expression simplifies to 19.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form of a^x, where 'a' is a positive constant and 'x' is a variable. They are characterized by their rapid growth or decay, depending on the base. Understanding how to manipulate these functions is crucial for evaluating expressions involving exponents, such as 8^(log8 19).

Logarithms

A logarithm is the inverse operation to exponentiation, answering the question: to what exponent must a base be raised to produce a given number? For example, log_b(a) = c means b^c = a. In the expression 8^(log8 19), the logarithm helps simplify the evaluation by converting the exponent into a more manageable form.

Change of Base Formula

The change of base formula allows the conversion of logarithms from one base to another, expressed as log_b(a) = log_k(a) / log_k(b) for any positive k. This is particularly useful when dealing with logarithmic expressions that may not be straightforward to evaluate. In this case, it helps in understanding how to manipulate the expression involving log8.