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. Identify the expression: ${\log }_{16}4$.
2. Recall the definition of a logarithm: ${\log }_{b}a=c$ means $b^c=a$.
3. Set up the equation: ${16}^x=4$.
4. Express 16 and 4 as powers of 2: $16=2^4$ and $4=2^2$.
5. Rewrite the equation: $(2^4{)}^x=2^2$ and solve for $x$ by equating the exponents.

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.