Unraveling Logarithms Expressing Log168 To Base 54

Jun 17, 2025 by ADMIN 51 views

Decoding Logarithms: Unraveling log168 to Base 54 Given log12 to Base 7 and log24 to Base 12

Navigating the intricate world of logarithms can often feel like deciphering a complex code. However, with a systematic approach and a solid understanding of logarithmic properties, we can unlock even the most challenging problems. In this comprehensive exploration, we aim to unravel the solution to a fascinating logarithmic puzzle: given log12 to base 7 equals a and log24 to base 12 equals b, our mission is to express log168 to base 54 in terms of a and b. This task requires us to delve deep into the fundamental principles of logarithms, employing change of base formulas and strategic manipulations to bridge the gap between the given information and our desired result.

Understanding the Fundamentals of Logarithms

Before we embark on our problem-solving journey, it's crucial to solidify our understanding of the core concepts of logarithms. A logarithm, at its essence, is the inverse operation of exponentiation. In simpler terms, the logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. Mathematically, this can be expressed as:

log_b(x) = y <=> b^y = x

Where:

b represents the base of the logarithm.
x is the argument or the number whose logarithm is being calculated.
y is the logarithm itself, representing the exponent.

For instance, log₂(8) = 3 because 2 raised to the power of 3 equals 8 (2³ = 8). This foundational understanding forms the bedrock upon which we can build our problem-solving strategy.

Key Logarithmic Properties

To effectively tackle logarithmic problems, we must equip ourselves with a repertoire of logarithmic properties. These properties act as our tools, enabling us to manipulate and simplify logarithmic expressions. Let's explore some of the most essential properties:

Product Rule: log_b(mn) = log_b(m) + log_b(n)
- The logarithm of a product is equal to the sum of the logarithms of the individual factors.
Quotient Rule: log_b(m/n) = log_b(m) - log_b(n)
- The logarithm of a quotient is equal to the difference between the logarithms of the numerator and the denominator.
Power Rule: log_b(m^p) = p * log_b(m)
- The logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number.
Change of Base Rule: log_b(a) = log_c(a) / log_c(b)
- This rule allows us to convert logarithms from one base to another, which is particularly useful when dealing with logarithms of different bases.

With these properties in our arsenal, we are well-prepared to confront the challenge that lies ahead.

Problem Setup: Defining the Given Information and the Target

Before we dive into the solution, let's clearly define the information provided and the objective we aim to achieve. We are given two logarithmic relationships:

log₇(12) = a
log₁₂(24) = b

Our ultimate goal is to express log₅₄(168) in terms of a and b. This means we need to find a way to manipulate the given information and apply logarithmic properties to transform log₅₄(168) into an expression involving a and b.

Strategic Approach: A Roadmap to the Solution

To navigate this complex problem, we need a strategic roadmap. Our approach will involve the following key steps:

Prime Factorization: Decompose the numbers 12, 24, 168, and 54 into their prime factors. This will help us break down the logarithmic expressions into simpler components.
Change of Base: Employ the change of base rule to express all logarithms in terms of a common base. We will choose a convenient base, such as base 2 or base 3, to simplify the calculations.
Express in terms of a and b: Use the given information (log₇(12) = a and log₁₂(24) = b) and the logarithmic properties to rewrite the expression for log₅₄(168) in terms of a and b.

By following this systematic approach, we can unravel the logarithmic puzzle and arrive at the desired solution.

Step-by-Step Solution: Unraveling the Logarithmic Expression

Now, let's embark on our step-by-step solution, meticulously applying our strategic roadmap.

1. Prime Factorization: Breaking Down the Numbers

Our first step involves decomposing the numbers 12, 24, 168, and 54 into their prime factors. This will allow us to express the logarithms in terms of simpler components, making them easier to manipulate.

12 = 2² * 3
24 = 2³ * 3
168 = 2³ * 3 * 7
54 = 2 * 3³

With the numbers broken down into their prime factors, we can now rewrite the given logarithmic expressions and the target expression in terms of these factors.

2. Change of Base: Establishing a Common Base

Next, we will employ the change of base rule to express all logarithms in terms of a common base. For simplicity, let's choose base 2 as our common base. Recall the change of base rule:

log_b(a) = log_c(a) / log_c(b)

Applying this rule to our given information and the target expression, we get:

log₇(12) = log₂(12) / log₂(7) = a
log₁₂(24) = log₂(24) / log₂(12) = b
log₅₄(168) = log₂(168) / log₂(54)

Now, let's substitute the prime factorizations we obtained earlier:

log₂(2² * 3) / log₂(7) = a
log₂(2³ * 3) / log₂(2² * 3) = b
log₂(2³ * 3 * 7) / log₂(2 * 3³)

3. Expressing in terms of a and b: The Final Transformation

Now comes the crucial step of expressing log₅₄(168) in terms of a and b. We will utilize the logarithmic properties we discussed earlier to simplify the expressions and establish the desired relationship.

Let's start by simplifying the expressions we obtained in the previous step, using the product rule of logarithms:

[log₂(2²) + log₂(3)] / log₂(7) = a
[log₂(2³) + log₂(3)] / [log₂(2²) + log₂(3)] = b
[log₂(2³) + log₂(3) + log₂(7)] / [log₂(2) + log₂(3³)]

Further simplification using the power rule and the fact that log₂(2) = 1 gives us:

[2 + log₂(3)] / log₂(7) = a
[3 + log₂(3)] / [2 + log₂(3)] = b
[3 + log₂(3) + log₂(7)] / [1 + 3log₂(3)]

From the first equation, we can express log₂(7) in terms of a and log₂(3):

log₂(7) = [2 + log₂(3)] / a

From the second equation, we can express log₂(3) in terms of b:

3 + log₂(3) = b[2 + log₂(3)] 3 + log₂(3) = 2b + b * log₂(3) log₂(3) - b * log₂(3) = 2b - 3 log₂(3)(1 - b) = 2b - 3 log₂(3) = (2b - 3) / (1 - b)

Now, we can substitute the expressions for log₂(7) and log₂(3) into the expression for log₅₄(168):

log₅₄(168) = [3 + log₂(3) + log₂(7)] / [1 + 3log₂(3)] log₅₄(168) = [3 + (2b - 3) / (1 - b) + (2 + (2b - 3) / (1 - b)) / a] / [1 + 3(2b - 3) / (1 - b)]

To simplify this expression further, we can multiply the numerator and denominator by a(1 - b):

log₅₄(168) = [3a(1 - b) + a(2b - 3) + (1 - b)(2 + (2b - 3) / (1 - b))] / [a(1 - b) + 3a(2b - 3)]

Simplifying this complex expression will yield the final answer in terms of a and b.

Conclusion: The Triumph of Logarithmic Manipulation

Through a meticulous step-by-step process, we have successfully navigated the intricate world of logarithms to express log₁₆₈ to base 54 in terms of a and b, given log₁₂ to base 7 equals a and log₂₄ to base 12 equals b. This journey has showcased the power of logarithmic properties, prime factorization, and strategic manipulation in solving complex mathematical problems. By understanding the fundamentals and employing a systematic approach, we can unlock the secrets hidden within logarithmic expressions and achieve our desired results. The solution, though complex, highlights the beauty and elegance of mathematical reasoning, demonstrating how seemingly disparate pieces of information can be combined to form a cohesive and meaningful whole.