Simplify Log 14 6 - Log 14 3/2 Express As Single Logarithm

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In the realm of mathematics, logarithms serve as powerful tools for simplifying complex calculations and revealing hidden relationships within exponential equations. Understanding the properties of logarithms is crucial for solving a wide range of mathematical problems, from basic algebraic manipulations to advanced calculus applications. This article delves into the fundamental properties of logarithms and demonstrates how they can be applied to simplify logarithmic expressions, focusing on the specific example of expressing log⁡146−log⁡1432\log_{14} 6 - \log_{14} \frac{3}{2} as a single logarithm. Through a step-by-step approach, we will unravel the intricacies of logarithmic operations and empower you to confidently tackle similar problems.

Logarithmic Fundamentals

Before we embark on the journey of simplifying the given expression, it is essential to establish a solid foundation in the basics of logarithms. A logarithm is essentially 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⁡ba=c\log_b a = c if and only if bc=ab^c = a

where:

  • bb is the base of the logarithm (a positive number not equal to 1)
  • aa is the argument of the logarithm (a positive number)
  • cc is the exponent or the logarithm itself

For instance, log⁡10100=2\log_{10} 100 = 2 because 102=10010^2 = 100. This signifies that the logarithm of 100 to the base 10 is 2, as 10 raised to the power of 2 equals 100. Grasping this fundamental relationship between logarithms and exponentiation is the cornerstone for comprehending the properties that govern logarithmic operations.

Key Logarithmic Properties

To effectively manipulate and simplify logarithmic expressions, we must familiarize ourselves with the fundamental properties of logarithms. These properties provide the rules for combining, expanding, and simplifying logarithmic terms. The key properties that are most relevant to our task include:

  1. Product Rule: The logarithm of the product of two numbers is equal to the sum of their logarithms. Mathematically, this is expressed as:

    log⁡b(mn)=log⁡bm+log⁡bn\log_b (mn) = \log_b m + \log_b n

  2. Quotient Rule: The logarithm of the quotient of two numbers is equal to the difference of their logarithms. Mathematically, this is expressed as:

    log⁡b(mn)=log⁡bm−log⁡bn\log_b (\frac{m}{n}) = \log_b m - \log_b n

  3. Power Rule: The logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number. Mathematically, this is expressed as:

    log⁡b(mp)=plog⁡bm\log_b (m^p) = p \log_b m

These properties act as the building blocks for simplifying logarithmic expressions. By understanding and applying these rules, we can effectively transform complex logarithmic expressions into simpler, more manageable forms. In the context of our given problem, the quotient rule will play a pivotal role in combining the two logarithmic terms into a single logarithm.

Applying Logarithmic Properties to Simplify the Expression

Now that we have a firm grasp of the fundamental properties of logarithms, we can proceed to simplify the given expression: log⁡146−log⁡1432\log_{14} 6 - \log_{14} \frac{3}{2}. Our goal is to express this difference of logarithms as a single logarithm. To achieve this, we will leverage the quotient rule of logarithms, which states that the logarithm of the quotient of two numbers is equal to the difference of their logarithms.

Step 1: Applying the Quotient Rule

The quotient rule allows us to rewrite the difference of logarithms as a single logarithm of a quotient. In our case, we have:

log⁡146−log⁡1432=log⁡14(632)\log_{14} 6 - \log_{14} \frac{3}{2} = \log_{14} (\frac{6}{\frac{3}{2}})

We have now successfully combined the two logarithmic terms into a single logarithm, with the argument being the quotient of 6 and 32\frac{3}{2}. The next step involves simplifying this complex fraction to obtain a more concise expression.

Step 2: Simplifying the Quotient

To simplify the quotient 632\frac{6}{\frac{3}{2}}, we need to divide 6 by the fraction 32\frac{3}{2}. Dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, we can rewrite the quotient as:

632=6×23\frac{6}{\frac{3}{2}} = 6 \times \frac{2}{3}

Now, we can perform the multiplication:

6×23=6×23=123=46 \times \frac{2}{3} = \frac{6 \times 2}{3} = \frac{12}{3} = 4

Thus, the simplified quotient is 4. We can now substitute this value back into our logarithmic expression.

Step 3: Expressing as a Single Logarithm

Substituting the simplified quotient back into our expression, we get:

log⁡14(632)=log⁡144\log_{14} (\frac{6}{\frac{3}{2}}) = \log_{14} 4

Therefore, the expression log⁡146−log⁡1432\log_{14} 6 - \log_{14} \frac{3}{2} can be expressed as a single logarithm, log⁡144\log_{14} 4.

Conclusion: Mastering Logarithmic Simplification

In this comprehensive exploration, we have successfully simplified the logarithmic expression log⁡146−log⁡1432\log_{14} 6 - \log_{14} \frac{3}{2} into a single logarithm, log⁡144\log_{14} 4. We achieved this by meticulously applying the quotient rule of logarithms and simplifying the resulting quotient. This process underscores the power of understanding and applying logarithmic properties to manipulate and simplify complex expressions.

By mastering the fundamental properties of logarithms, you equip yourself with a powerful toolkit for tackling a wide array of mathematical problems. Whether you are solving exponential equations, analyzing financial data, or exploring scientific phenomena, the ability to manipulate logarithmic expressions is an invaluable asset. Embrace the power of logarithms and unlock new dimensions of mathematical understanding.

Key takeaways:

  • Logarithms are the inverse of exponential functions.
  • The quotient rule of logarithms allows us to combine the difference of logarithms into a single logarithm of a quotient.
  • Simplifying fractions is crucial for obtaining the final simplified logarithmic expression.

By consistently practicing and applying these principles, you can confidently navigate the world of logarithms and harness their power to solve complex problems.

Further Practice

To solidify your understanding of logarithmic simplification, try applying the same principles to solve similar problems. For instance, consider simplifying expressions such as:

  • log⁡510−log⁡52\log_5 10 - \log_5 2
  • log⁡224−log⁡23\log_2 24 - \log_2 3
  • log⁡318−log⁡32\log_3 18 - \log_3 2

By working through these examples, you will further refine your skills and develop a deeper appreciation for the elegance and power of logarithms.

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