Problem 40 - Intermediate Algebra

Question

For Problems \(21-40\), evaluate each expression. $$ \log _{2}\left(\log _{5} 5\right) $$

Step-by-Step Solution

Answer

0

Step 1: Evaluate the Innermost Logarithm

The expression to evaluate is \( \log _{2} \left( \log _{5} 5 \right) \). Start by evaluating the innermost logarithmic term, \( \log _{5} 5 \). By the basic logarithmic identity, \( \log_b b = 1 \), since any number raised to the power of 1 is itself. Therefore, \( \log_5 5 = 1 \).

Step 2: Evaluate the Outer Logarithm

With the innermost logarithm evaluated, the expression now becomes \( \log_2(1) \). Now evaluate \( \log_2(1) \). Remember that \( \log_b(1) = 0 \) for any base \( b > 0 \) and \( b eq 1 \), because any number to the power of 0 is 1. Therefore, \( \log_2(1) = 0 \).

Key Concepts

Logarithmic Identities

Logarithmic identities are essential tools that simplify solving logarithmic expressions. These identities help us understand the inherent relationships within logarithms, making complex problems much easier to handle.

One primary identity every student should know is the Power Identity: \( \log_b(b) = 1 \). This identity tells us that any logarithm of a number when the base is the same as the number itself equals 1. For instance, \( \log_5(5) = 1 \). This is because any base raised to the power of 1 will always equal the base itself.

Another vital identity is the Zero Identity: \( \log_b(1) = 0 \). No matter the base (as long as it's greater than zero and not equal to one), the logarithm of 1 will always be 0. This results from the mathematical principle that any number raised to the power of zero is 1. By understanding these identities, you can quickly simplify many logarithmic expressions, which is crucial for solving problems efficiently.

Nested Logarithms

Nested logarithms might seem complex at first, but they become manageable once you recognize that you should work from the inside out. Let's break down how to manage them effectively.

When dealing with nested logarithms, such as \( \log_2(\log_5 5) \), the key is to simplify the innermost logarithm first. In this example:

• The innermost logarithm is \( \log_5 5 \).
• By applying the Power Identity, we see that \( \log_5 5 = 1 \).

After simplifying the inside, the expression transforms to \( \log_2(1) \). From there, use the Zero Identity to find that any logarithm of 1 is 0. Therefore, \( \log_2(1) = 0 \).

Nesting is just a way of piling operations inside each other—simplify each layer one at a time, and the whole problem becomes easy to tackle.

Evaluating Expressions

When evaluating expressions that include logarithms, such as the one given, it's crucial to approach it with a step-by-step strategy. Here are some strategies to help break down such problems efficiently:

• Begin with identifying and using logarithmic identities to simplify each part of the expression.
• Evaluate from the innermost part of the expression outwards, as is common with nested logarithms.

In the provided problem, we first looked at the innermost expression \( \log_5 5 \) and used the Power Identity to reduce it to 1. Then, we applied this simplified expression to the outer operation, \( \log_2(1) \), and found the result using the Zero Identity.

By using clear and systematic methods, evaluating logarithmic expressions not only becomes more straightforward but also helps reinforce the understanding of foundational logarithmic concepts.