Simplifying Logarithmic Expressions: A Step-by-Step Guide

Dec 2, 2025 by Alex Johnson 58 views

Have you ever encountered a complex logarithmic expression and felt overwhelmed? Don't worry, you're not alone! Logarithms can seem intimidating at first, but with a clear understanding of the rules and properties, you can simplify even the most intricate expressions. In this guide, we'll break down the process of expressing a given logarithmic expression as a single logarithm with a coefficient of 1, simplifying it along the way. We'll focus on the specific example:

$\frac{1}{2}\left[14 \ln (d+3)+\ln d-\ln d^2\right]=\square$

So, let's dive in and conquer the world of logarithms together!

Understanding Logarithmic Properties

Before we tackle the main problem, it's crucial to have a solid grasp of the fundamental logarithmic properties. These properties are the tools we'll use to manipulate and simplify the expression. Let's review the key ones:

Power Rule: This rule states that ${\log_b(x^p) = p \log_b(x)}$ . In simpler terms, if you have an exponent inside a logarithm, you can move it to the front as a coefficient.
Product Rule: The product rule says that ${\log_b(xy) = \log_b(x) + \log_b(y)}$ . This means the logarithm of a product is the sum of the logarithms of the individual factors.
Quotient Rule: Conversely, the quotient rule states that ${\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)}$ . The logarithm of a quotient is the difference between the logarithms of the numerator and the denominator.
Coefficient Rule (Reverse Power Rule): This is simply the power rule in reverse: ${p \log_b(x) = \log_b(x^p)}$ . We can move a coefficient in front of a logarithm back into the logarithm as an exponent.

These four properties are our main weapons in simplifying logarithmic expressions. We will apply these rules strategically to combine and condense the terms in our expression.

Step-by-Step Simplification

Now that we've refreshed our understanding of logarithmic properties, let's apply them to simplify the expression:

$\frac{1}{2}\left[14 \ln (d+3)+\ln d-\ln d^2\right]$

Our goal is to express this as a single logarithm with a coefficient of 1. Here's how we'll do it:

1. Apply the Reverse Power Rule

First, we'll use the reverse of the power rule to move the coefficient 14 inside the first logarithm:

$\frac{1}{2}\left[\ln (d+3)^{14} + \ln d - \ln d^2\right]$

This step helps us consolidate the terms within the brackets. Now we have exponents inside the logarithms, making it easier to apply the product and quotient rules.

2. Apply the Product Rule

Next, we'll use the product rule to combine the first two logarithmic terms. Remember, ${\ln a + \ln b = \ln(ab)}$ . So, we have:

$\frac{1}{2}\left[\ln ((d+3)^{14} \cdot d) - \ln d^2\right]$

We've now combined the sum of the first two logarithms into a single logarithm of a product. The expression is becoming more compact.

3. Apply the Quotient Rule

Now, we'll use the quotient rule to combine the remaining terms. Recall that ${\ln a - \ln b = \ln(\frac{a}{b})}$ . Applying this rule, we get:

$\frac{1}{2}\left[\ln \left(\frac{(d+3)^{14} \cdot d}{d^2}\right)\right]$

We've successfully combined all the logarithms inside the brackets into a single logarithm of a quotient. Now we need to simplify the fraction inside the logarithm.

4. Simplify the Expression Inside the Logarithm

Let's simplify the fraction ${\frac{(d+3)^{14} \cdot d}{d^2}}$ . We can cancel out one factor of d from the numerator and the denominator:

$\frac{(d+3)^{14} \cdot d}{d^2} = \frac{(d+3)^{14}}{d}$

So, our expression now looks like:

$\frac{1}{2}\left[\ln \left(\frac{(d+3)^{14}}{d}\right)\right]$

5. Apply the Power Rule (Again)

Finally, we need to deal with the coefficient of ${\frac{1}{2}}$ outside the logarithm. We'll use the power rule again, but this time we'll move the coefficient inside the logarithm as an exponent:

$\ln \left(\left(\frac{(d+3)^{14}}{d}\right)^{\frac{1}{2}}\right)$

6. Simplify the Exponent

Raising something to the power of ${\frac{1}{2}}$ is the same as taking the square root. So we can rewrite the expression as:

$\ln \left(\sqrt{\frac{(d+3)^{14}}{d}}\right)$

We can further simplify this by taking the square root of ${(d+3)^{14}}$ , which is ${(d+3)^7}$ . Our final simplified expression is:

$\ln \left(\frac{(d+3)^7}{\sqrt{d}}\right)$

Final Answer

Therefore, the logarithmic expression ${\frac{1}{2}\left[14 \ln (d+3)+\ln d-\ln d^2\right]}$ can be written as a single logarithm with a coefficient of 1 and simplified to:

$\ln \left(\frac{(d+3)^7}{\sqrt{d}}\right)$

Key Takeaways

Simplifying logarithmic expressions involves a systematic application of logarithmic properties. Here are the key takeaways from this example:

Master the Logarithmic Properties: Understanding the power, product, and quotient rules is essential.
Apply the Rules Strategically: Move coefficients as exponents, combine sums into products, and differences into quotients.
Simplify as You Go: Don't wait until the end to simplify. Simplify fractions and exponents as you proceed.
Practice Makes Perfect: The more you practice, the more comfortable you'll become with manipulating logarithmic expressions.

Tips for Success

Here are some extra tips to help you succeed in simplifying logarithmic expressions:

Write Clearly: Keep your work organized and easy to follow. This will help you avoid mistakes.
Double-Check Your Work: Logarithmic properties can be tricky, so always double-check each step.
Don't Be Afraid to Break It Down: If the expression seems overwhelming, break it down into smaller steps.
Use Parentheses: Use parentheses to keep track of what operations you've performed.
Look for Opportunities to Simplify: Always be on the lookout for ways to simplify the expression further.

By following these tips and practicing regularly, you'll be able to tackle any logarithmic expression with confidence. Remember, the key is to understand the properties and apply them systematically.

Conclusion

Simplifying logarithmic expressions is a valuable skill in mathematics. By understanding and applying the fundamental logarithmic properties, you can transform complex expressions into simpler, more manageable forms. We've walked through a detailed example, breaking down each step and explaining the reasoning behind it. With practice and patience, you can master the art of simplifying logarithms and unlock their power in various mathematical contexts. Keep practicing, and you'll become a logarithm pro in no time!

For further learning and practice, you can explore resources like Khan Academy's Logarithm Section. This can provide additional examples, exercises, and explanations to solidify your understanding.