Problem 132 - Elementary and Intermediate Algebra

Question

Evaluate: \(\log _{6}\left(\log _{5}\left(\log _{4} 1,024\right)\right)\)

Step-by-Step Solution

Answer

The value of the expression is 0.

Step 1: Simplify the Logarithm of 1024 with Base 4

We know that 1024 can be expressed as a power of 2. Since \(1024 = 2^{10}\), let's express it with base 4: \(1024 = (2^2)^5 = 4^5\). Therefore, \(\log_4 1024 = 5\) because \(4^5 = 1024\). This simplifies the original expression to \(\log_6(\log_5(5))\).

Step 2: Evaluate the Logarithm with Base 5

At this point, we need to evaluate \(\log_5(5)\). We know that if the base of the logarithm and the argument are the same, the answer is 1. Therefore, \(\log_5(5) = 1\). This simplifies our expression further to \(\log_6(1)\).

Step 3: Evaluate the Logarithm with Base 6

Now we need to evaluate the expression \(\log_6(1)\). We know that \(\log_b(1) = 0\) for any base \(b\) because any number raised to the power 0 is 1. Therefore, \(\log_6(1) = 0\). This is the final value of the original expression.

Key Concepts

Logarithms

Logarithms are an essential mathematical tool that helps us solve exponential equations by reversing the process of exponentiation. Simply put, a logarithm answers the question: "To what power must the base be raised, to produce a given number?" For instance, if we have \(\log_{b}(x) = y\), it implies that if we raise the base \(b\) to the power \(y\) we will get \(x\), hence, \(b^y = x\).

• For example, \(\log_{10}(100) = 2\) because \(10^2 = 100\).
• Another example is \(\log_{2}(8) = 3\) because \(2^3 = 8\).

A special case of logarithms occurs when the argument equals the base, resulting in a value of 1. That's because any number raised to the power of 1 equals itself, so \(\log_{b}(b) = 1\). Additionally, \(\log_b(1) = 0\) proves that the logarithm of 1 in any base is zero since any number raised to the power of zero gives 1.

Base Conversion

Understanding how to work with different bases is a crucial part of simplifying logarithmic expressions. Often, converting expressions into the same base or recognizing powers of numbers can make the process much easier.
In the problem given, \(\log _{4} 1024\) was simplified by first expressing 1024 as \(2^{10}\) since 1024 is a power of 2. This process led to recognizing that:

• \(1024 = (2^2)^5 = 4^5\), meaning \(1024\) can be represented as \(4^5\).
• Thus, \(\log_4 1024 = 5\) because the logarithm asks what power 4 needs to be raised to, to get 1024, and the answer is 5.

Recognizing and converting between bases can certainly demystify logs and make complex expressions simpler. It's also worth mentioning that various rules like the change of base formula can assist in converting between bases when required. However, in this problem, direct simplification was used since it made the steps straightforward.

Step-by-step Solutions

Approaching a problem step-by-step is key to tackling complex expressions, especially with logarithms and different bases. Here's how a systematic solution could look for evaluating the nested logarithmic problem:

• Step 1: Start with the innermost logarithm. For \(1024\), rewrite it in a way that relates to the base. Here, recognize it's \(4^5\), allowing you to easily simplify \(\log_4 1024\) to 5.
• Step 2: Move onto the next layer in the nested log expression. With \(\log_5(5)\), remember that when the base matches the argument, the result is always 1. So, \(\log_5(5) = 1\).
• Step 3: Finally, address the remaining log. Since \(\log_6(1) = 0\) for any base, you deduce the final result is 0.

Breaking it down ensures accuracy and improves understanding. When doubting what rules apply, always go back to the definitions, such as any log with an argument of 1 equals 0, or a matching base and argument equals 1.