Problem 38 - Intermediate Algebra

Question

For Problems \(21-40\), evaluate each expression. $$ \log _{2}\left(\log _{4} 16\right) $$

Step-by-Step Solution

Answer

1

Step 1: Evaluate the Inner Logarithm

We start by evaluating the inner logarithm, which is \( \log_4 16 \). To do this, remember that \( 4^x = 16 \). We need to find \( x \) such that this is true. Since \( 4^2 = 16 \), we have \( \log_4 16 = 2 \).

Step 2: Substitute into the Outer Logarithm

Now that we have \( \log_4 16 = 2 \), substitute this into the outer logarithm: \( \log_2 2 \).

Step 3: Evaluate the Outer Logarithm

Evaluate \( \log_2 2 \). Since \( 2^1 = 2 \), \( \log_2 2 = 1 \). This completes the evaluation.

Key Concepts

Evaluating Logarithms

Evaluating logarithms is an essential skill in mathematics that involves determining the power to which a base number must be raised to get a certain value. To evaluate a logarithm like \( \log_b a \), we consider the question: "What power must \( b \) be raised to in order to obtain \( a \)?" This power is the logarithm.
To illustrate, in our example, the inner expression \( \log_4 16 \) is asking for the power needed to raise 4 to obtain 16.
Recognizing that \( 4^2 = 16 \), we can conclude that \( \log_4 16 = 2 \). This step essentially translates a logarithmic expression into an equation that can be solved much like any other algebraic expression.
This approach of converting to exponential form streamlines the process of solving logarithmic expressions.

Nested Logarithmic Expressions

Nested logarithmic expressions involve solving logarithms that are contained within other logarithms. This can seem daunting at first, but the method is straightforward. We break it down into smaller, more manageable parts.
In our example \( \log_2(\log_4 16) \), the expression is nested because the evaluation of \( \log_4 16 \) results in a value that is used as the argument for another logarithm, \( \log_2 \).
By tackling the innermost logarithm first, we simplify the expression step-by-step. Once the innermost value is found, it’s used in the next outer layer of calculations. Thus, solving nested logarithms is akin to peeling an onion—layer by layer, ensuring that at each stage correct simplifications are applied before progressing to the next layer.

Step-by-Step Math Solutions

Step-by-step math solutions are invaluable when dealing with complex expressions. They provide clarity by breaking a problem into more manageable pieces, ensuring that each component is correctly understood and solved before moving on.
In our example, the process is divided into distinct parts: evaluating \( \log_4 16 \), substituting this into the outer logarithm, and finally evaluating \( \log_2 2 \).
This systematic approach ensures that errors are minimized and each logical step from the inner to outermost expressions is clear.

• First, evaluate the innermost expression.
• Next, substitute this result into the outer expression.
• Finally, solve the outer logarithm as a separate step.

The step-by-step technique is not just a methodology; it’s a way to cultivate deeper understanding and mastery over mathematical techniques.