Question

The value of \(\log _{3}\left[\log _{2}\left\\{\log _{4}\left(\log _{5} 625^{4}\right)\right\\}\right]\) is. (1) 0 (2) 1 (3) 2 (4) \(\log _{3} 4\)

Short answer

Answer: (1) 0

Step-by-step solution

Simplify the Innermost Logarithm: \(\log_{5}(625^4)\)

First, we'll simplify the innermost logarithm. Using the power rule for logarithms, this expression can be written as: \(\log_{5}(625^4) = 4 \times \log_{5}(625)\) Since \(625=5^4\), we can write the expression as: \(4 \times \log_{5}(5^4)\) Now, applying the inverse property of logarithms, we get: \(4 \times 4 = 16\)

Simplify the Second Logarithm: \(\log_{4}(16)\)

Now, we will simplify the second logarithm: \(\log_{4}(16)\) Since \(16=4^2\), we can write the expression as: \(\log_{4}(4^2)\) Applying the inverse property of logarithms, we get: \(2\)

Simplify the Third Logarithm: \(\log_{2}(2)\)

Now, we will simplify the third logarithm: \(\log_{2}(2)\) Since \(2=2^1\), we can write the expression as: \(\log_{2}(2^1)\) Applying the inverse property of logarithms, we get: \(1\)

Simplify the Outermost Logarithm: \(\log_{3}(1)\)

Finally, we will simplify the outermost logarithm: \(\log_{3}(1)\) Since \(1=3^0\), we can write the expression as: \(\log_{3}(3^0)\) Applying the inverse property of logarithms, we get: \(0\) Thus, the value of \(\log _{3}\left[\log _{2}\left\{\log _{4}\left(\log _{5}(625^{4})\right)\right\}\right]\) is 0. The correct answer is (1) 0.

Key concepts

Power Rule for Logarithms

The power rule for logarithms is a helpful tool in mathematics. It allows you to simplify expressions where the argument of the logarithm is raised to a power. This rule states that:
\[\log_b(a^n) = n \cdot \log_b(a)\]
In other words, the exponent can be brought as a coefficient in front of the logarithm. This makes computations easier by turning complex products into simpler additions.
In the original exercise, we applied this rule to the innermost expression:
\(\log_{5}(625^4) = 4 \cdot \log_{5}(625)\).
By breaking down the power, we begin simplifying the nested logarithmic expression efficiently.

Inverse Property of Logarithms

The inverse property of logarithms is a fundamental concept that helps in simplifying expressions where the base of the logarithm and the argument are the same.
This property is expressed as:
\[\log_b(b^n) = n\]
Essentially, if you have a logarithm with the same number as the base and the argument, the result is simply the exponent.In our exercise, the inverse property of logarithms was repeatedly used. For instance, after applying the power rule, we had:
\(4 \cdot \log_{5}(5^4)\).
By applying the inverse property, we found \(4 \cdot 4 = 16\),simplifying the argument further. This principle was crucial at each stage of the problem-solving process.

Simplifying Nested Logarithms

Dealing with nested logarithms can seem daunting, but understanding the procedures makes it manageable. The concept involves simplifying from the innermost expression outwards, much like peeling an onion.The original problem was:
\(\log_{3}\left[\log_{2}\left\{\log_{4}\left(\log_{5}(625^4)\right)\right\}\right]\)
By simplifying step by step:

• Begin with \(\log_{5}(625^4)\).
• Once simplified to a single number, move to the outer layer \(\log_{4}(16)\).
• Continuing outward to \(\log_{2}(2)\).
• Finally, solving the outermost layer \(\log_{3}(1)\).

Each layer builds on the last, gradually reducing complexity until the entire expression is simplified. This is a key strategic approach in mathematical problem-solving.

Mathematics Problem Solving

Mathematics problem solving is often about strategy and a step-by-step approach. A clear method can transform seemingly intricate problems into simpler ones, just like the nested logarithms problem. To tackle problems effectively:

• Understand and isolate the innermost expressions first.
• Apply rules systematically, such as the power and inverse properties of logarithms.
• Learn to break down the problem, addressing each part individually before proceeding to the next.

This methodical approach not only simplifies the solution process but enhances understanding. Mastering these strategies builds a strong mathematical foundation, enabling you to tackle a variety of problems with confidence.