In Exercises 21–42, evaluate each expression without using a calculator. log4 16

Verified step by step guidance

1. Recognize that the expression ${\log }_{4}16$ is asking for the power to which 4 must be raised to get 16.
2. Express 16 as a power of 4. Notice that 16 can be written as $4^2$.
3. Rewrite the logarithmic expression using the equivalent exponential form: $4^x=16$.
4. Substitute the expression for 16: $4^x=4^2$.
5. Since the bases are the same, set the exponents equal to each other: $x=2$.

Key Concepts

Logarithms

A logarithm is the power to which a base must be raised to obtain a given number. In the expression log_b(a), 'b' is the base, 'a' is the number, and the result is the exponent 'x' such that b^x = a. Understanding logarithms is essential for solving problems involving exponential relationships.

Change of Base Formula

The change of base formula allows you to convert logarithms from one base to another. It states that log_b(a) can be expressed as log_k(a) / log_k(b) for any positive base 'k'. This is particularly useful when the base is not easily computable or when using a calculator with limited base options.

Exponential Equations

Exponential equations involve expressions where variables appear as exponents. To evaluate logarithmic expressions, it is often helpful to rewrite them in exponential form. For example, log4(16) can be interpreted as finding the exponent 'x' such that 4^x = 16, which simplifies the evaluation process.