In Exercises 19–29, evaluate each expression without using a calculator. If evaluation is not possible, state the reason. log16 4

Verified step by step guidance

1. Rewrite the logarithmic expression using the definition of a logarithm: \( \log_b a = c \) means \( b^c = a \). Here, \( \log_{16} 4 \) implies finding the value of \( c \) such that \( 16^c = 4 \).
2. Express the base (16) and the result (4) as powers of the same base. Note that \( 16 = 2^4 \) and \( 4 = 2^2 \).
3. Substitute these expressions into the equation \( 16^c = 4 \). This becomes \( (2^4)^c = 2^2 \).
4. Simplify the left-hand side using the power rule \( (a^m)^n = a^{m \cdot n} \). This gives \( 2^{4c} = 2^2 \).
5. Since the bases are the same, set the exponents equal to each other: \( 4c = 2 \). Solve for \( c \) by dividing both sides by 4: \( c = \frac{2}{4} \).

Key Concepts

Logarithms

A logarithm is the inverse operation to exponentiation, representing the power to which a base must be raised to obtain a given number. For example, in the expression log_b(a), b is the base, and the result is the exponent x such that b^x = a. Understanding logarithms is essential for evaluating expressions like log16 4.

Change of Base Formula

The change of base formula allows us to convert logarithms from one base to another, making them easier to evaluate. It states that log_b(a) can be expressed as log_k(a) / log_k(b) for any positive k. This is particularly useful when the base is not a common one, such as in log16 4, where we can convert to base 2 or 10.

Properties of Logarithms

Logarithms have several properties that simplify their evaluation, including the product, quotient, and power rules. For instance, log_b(mn) = log_b(m) + log_b(n) and log_b(m/n) = log_b(m) - log_b(n). These properties can help break down complex logarithmic expressions into simpler components, aiding in their evaluation.