Condense The Logarithmic Expression.$\log 15 - (2 \log X + 3 \log Y + \log 7$\]

Introduction

Logarithmic expressions can be complex and difficult to work with, especially when it comes to simplifying them. In this article, we will focus on condensing logarithmic expressions, specifically the expression $\log 15 - (2 \log x + 3 \log y + \log 7)$. We will break down the steps involved in condensing this expression and provide a clear understanding of the process.

Understanding Logarithmic Properties

Before we dive into condensing the expression, it's essential to understand the properties of logarithms. The three main properties of logarithms are:

• Product Property: $\log (a \cdot b) = \log a + \log b$
• Quotient Property: $\log \left(\frac{a}{b}\right) = \log a - \log b$
• Power Property: $\log (a^b) = b \cdot \log a$

These properties will be crucial in simplifying the given expression.

Condensing the Expression

Now that we have a good understanding of logarithmic properties, let's focus on condensing the expression $\log 15 - (2 \log x + 3 \log y + \log 7)$.

Step 1: Apply the Distributive Property

The first step in condensing the expression is to apply the distributive property. This means that we need to distribute the negative sign to each term inside the parentheses.

$\log 15 - (2 \log x + 3 \log y + \log 7) = \log 15 - 2 \log x - 3 \log y - \log 7$

Step 2: Combine Like Terms

Now that we have distributed the negative sign, we can combine like terms. In this case, we have two logarithmic terms with the same base, which are $\log 15$ and $-\log 7$. We can combine these terms using the quotient property.

$\log 15 - 2 \log x - 3 \log y - \log 7 = \log \left(\frac{15}{7}\right) - 2 \log x - 3 \log y$

Step 3: Apply the Power Property

The next step is to apply the power property to the terms $2 \log x$ and $3 \log y$. This means that we need to multiply the coefficient of the logarithm by the logarithm itself.

$\log \left(\frac{15}{7}\right) - 2 \log x - 3 \log y = \log \left(\frac{15}{7}\right) - \log (x^2) - \log (y^3)$

Step 4: Combine Like Terms Again

Now that we have applied the power property, we can combine like terms again. In this case, we have two logarithmic terms with the same base, which are $\log \left(\frac{15}{7}\right)$ and $-\log (x^2)$. We can combine these terms using the quotient property.

$\log \left(\frac{15}{7}\right) - \log (x^2) - \log (y^3) = \log \left(\frac{\frac{15}{7}}{x^2}\right) - \log (y^3)$

Step 5: Simplify the Expression

Finally, we can simplify the expression by combining the two logarithmic terms.

$\log \left(\frac{\frac{15}{7}}{x^2}\right) - \log (y^3) = \log \left(\frac{\frac{15}{7}}{x^2 y^3}\right)$

Conclusion

Condensing logarithmic expressions can be a complex process, but by understanding the properties of logarithms and following the steps outlined in this article, we can simplify even the most complex expressions. In this case, we were able to condense the expression $\log 15 - (2 \log x + 3 \log y + \log 7)$ into a single logarithmic term.

Common Mistakes to Avoid

When condensing logarithmic expressions, there are several common mistakes to avoid:

• Not applying the distributive property: Failing to distribute the negative sign to each term inside the parentheses can lead to incorrect results.
• Not combining like terms: Failing to combine like terms can make the expression more complex than it needs to be.
• Not applying the power property: Failing to apply the power property can make it difficult to simplify the expression.

Real-World Applications

Condensing logarithmic expressions has several real-world applications, including:

• Engineering: Logarithmic expressions are commonly used in engineering to describe complex systems and processes.
• Finance: Logarithmic expressions are used in finance to calculate interest rates and investment returns.
• Science: Logarithmic expressions are used in science to describe complex phenomena and processes.

Final Thoughts

Condensing logarithmic expressions is a complex process that requires a deep understanding of logarithmic properties and a step-by-step approach. By following the steps outlined in this article, we can simplify even the most complex expressions and gain a deeper understanding of logarithmic functions.