Evaluating Logarithmic Expressions If Ln A=2, Ln B=3, And Ln C=5

This article delves into solving logarithmic expressions given the values of ${\ln a}$, ${\ln b}$, and ${\ln c}$. We will explore how to use the properties of logarithms to evaluate more complex expressions. Specifically, we will solve two problems:

(a) Evaluate ${\ln \left(\frac{a^3}{b^4 c^4}\right)}$

(b) Evaluate ${\ln \sqrt{a^3 b^1 c^1}}$

Let's dive into the solutions step by step.

(a) Evaluating ${\ln \left(\frac{a^3}{b^4 c^4}\right)}$

To evaluate this expression, we'll use the properties of logarithms, which allow us to simplify complex logarithmic expressions into simpler terms. The key properties we will use are:

1. Quotient Rule: ${\ln \left(\frac{x}{y}\right) = \ln x - \ln y}$
2. Product Rule: ${\ln (xy) = \ln x + \ln y}$
3. Power Rule: ${\ln (x^p) = p \ln x}$

Given the expression ${\ln \left(\frac{a^3}{b^4 c^4}\right)}$, we can first apply the quotient rule to separate the numerator and the denominator:

${\ln \left(\frac{a^3}{b^4 c^4}\right) = \ln (a^3) - \ln (b^4 c^4)}$

Next, we apply the product rule to the term ${\ln (b^4 c^4)}$. This will separate the product into a sum of logarithms:

${\ln (b^4 c^4) = \ln (b^4) + \ln (c^4)}$

Substitute this back into the original expression:

${\ln (a^3) - \ln (b^4 c^4) = \ln (a^3) - [\ln (b^4) + \ln (c^4)]}$

Distribute the negative sign:

${\ln (a^3) - \ln (b^4) - \ln (c^4)}$

Now, we apply the power rule to each term. The power rule states that ${\ln (x^p) = p \ln x}$. Applying this to each term, we get:

${\ln (a^3) = 3 \ln a}$

${\ln (b^4) = 4 \ln b}$

${\ln (c^4) = 4 \ln c}$

Substitute these back into the expression:

${3 \ln a - 4 \ln b - 4 \ln c}$

We are given that ${\ln a = 2}$, ${\ln b = 3}$, and ${\ln c = 5}$. Substitute these values into the expression:

${3(2) - 4(3) - 4(5)}$

Perform the multiplication:

${6 - 12 - 20}$

Finally, perform the subtraction:

${6 - 12 - 20 = 6 - 32 = -26}$

Therefore, ${\ln \left(\frac{a^3}{b^4 c^4}\right) = -26}$.

This detailed step-by-step solution illustrates the application of logarithmic properties to simplify and evaluate the given expression. Understanding and applying these properties is crucial for solving more complex logarithmic problems. The use of the quotient, product, and power rules allows us to break down the initial expression into manageable parts, making the evaluation straightforward once the values of ${\ln a}$, ${\ln b}$, and ${\ln c}$ are substituted. By meticulously following each step, we arrive at the final answer of -26, demonstrating the power and elegance of logarithmic simplification techniques. Furthermore, this approach highlights the importance of mastering fundamental logarithmic identities and their application in problem-solving scenarios. The ability to manipulate logarithmic expressions is a valuable skill in various fields, including mathematics, physics, and engineering, where logarithmic scales and relationships are frequently encountered. Thus, understanding and practicing these techniques is essential for anyone seeking to excel in these areas. In conclusion, the solution not only provides the answer but also reinforces the understanding of logarithmic principles and their practical application.

(b) Evaluating ${\ln \sqrt{a^3 b^1 c^1}}$

To evaluate ${\ln \sqrt{a^3 b^1 c^1}}$, we again rely on the properties of logarithms, along with the understanding of how to express square roots as fractional exponents. The key properties we will use are:

1. Power Rule: ${\ln (x^p) = p \ln x}$
2. Product Rule: ${\ln (xy) = \ln x + \ln y}$

First, we express the square root as a fractional exponent. Recall that ${\sqrt{x} = x^{\frac{1}{2}}}$. Therefore, we can rewrite the expression as:

${\ln \sqrt{a^3 b^1 c^1} = \ln (a^3 b^1 c^1)^{\frac{1}{2}}}$

Next, we apply the power rule to the entire term. The power rule states that ${\ln (x^p) = p \ln x}$. Applying this, we get:

${\ln (a^3 b^1 c^1)^{\frac{1}{2}} = \frac{1}{2} \ln (a^3 b^1 c^1)}$

Now, we apply the product rule to the term ${\ln (a^3 b^1 c^1)}$. The product rule states that ${\ln (xy) = \ln x + \ln y}$. Applying this, we get:

${\ln (a^3 b^1 c^1) = \ln (a^3) + \ln (b^1) + \ln (c^1)}$

Substitute this back into the expression:

${\frac{1}{2} \ln (a^3 b^1 c^1) = \frac{1}{2} [\ln (a^3) + \ln (b^1) + \ln (c^1)]}$

Distribute the ${\frac{1}{2}}$:

${\frac{1}{2} \ln (a^3) + \frac{1}{2} \ln (b^1) + \frac{1}{2} \ln (c^1)}$

Apply the power rule again to each term:

${\frac{1}{2} \ln (a^3) = \frac{1}{2} (3 \ln a) = \frac{3}{2} \ln a}$

${\frac{1}{2} \ln (b^1) = \frac{1}{2} (1 \ln b) = \frac{1}{2} \ln b}$

${\frac{1}{2} \ln (c^1) = \frac{1}{2} (1 \ln c) = \frac{1}{2} \ln c}$

Substitute these back into the expression:

${\frac{3}{2} \ln a + \frac{1}{2} \ln b + \frac{1}{2} \ln c}$

We are given that ${\ln a = 2}$, ${\ln b = 3}$, and ${\ln c = 5}$. Substitute these values into the expression:

${\frac{3}{2} (2) + \frac{1}{2} (3) + \frac{1}{2} (5)}$

Perform the multiplication:

${3 + \frac{3}{2} + \frac{5}{2}}$

Combine the fractions:

${3 + \frac{3 + 5}{2} = 3 + \frac{8}{2}}$

Simplify:

${3 + 4 = 7}$

Therefore, ${\ln \sqrt{a^3 b^1 c^1} = 7}$.

This detailed solution illustrates how to handle logarithmic expressions involving square roots and products. The crucial step is recognizing the square root as a fractional exponent, which allows us to apply the power rule effectively. By breaking down the expression step by step and applying the properties of logarithms systematically, we can simplify and evaluate the given expression. The ability to convert radicals to exponents and apply the power and product rules of logarithms is fundamental in solving such problems. The final answer of 7 highlights the importance of careful manipulation and substitution to arrive at the correct result. Moreover, this approach reinforces the understanding of how different logarithmic properties interact and how they can be used in combination to solve complex problems. This skill is particularly valuable in fields such as calculus and differential equations, where logarithmic functions and their properties are frequently used. In summary, the solution not only provides the final answer but also serves as a comprehensive guide to applying logarithmic principles in problem-solving.

Conclusion

In this article, we have demonstrated how to evaluate logarithmic expressions using the properties of logarithms. We tackled two problems:

(a) ${\ln \left(\frac{a^3}{b^4 c^4}\right) = -26}$

(b) ${\ln \sqrt{a^3 b^1 c^1} = 7}$

By applying the quotient, product, and power rules of logarithms, we successfully simplified and evaluated the expressions. These properties are essential tools in dealing with logarithmic functions, and understanding them is crucial for solving various mathematical problems. Mastering these techniques not only helps in academic settings but also in practical applications where logarithmic scales are used, such as in scientific and engineering calculations.