Expanding Logarithmic Expressions: A Step-by-Step Guide

Nov 2, 2025 by SLV Team 56 views

In the realm of mathematics, particularly when dealing with logarithms, it's often necessary to expand a complex logarithmic expression into simpler terms involving sums and differences of individual logarithms. This process leverages several key properties of logarithms, making it easier to manipulate and solve equations or simplify expressions. This guide will walk you through the process of expanding the logarithmic expression $\log _c\left(\sqrt[4]{\frac{m^{16} n^{16}}{c^3 a^5}}\right)$ , assuming all variables are positive.

Understanding the Properties of Logarithms

Before we dive into the expansion, let's quickly recap the essential properties of logarithms that we'll be using:

Product Rule: $\log_b(xy) = \log_b(x) + \log_b(y)$ - The logarithm of a product is the sum of the logarithms.
Quotient Rule: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$ - The logarithm of a quotient is the difference of the logarithms.
Power Rule: $\log_b(x^p) = p \log_b(x)$ - The logarithm of a number raised to a power is the power times the logarithm of the number.
Root Rule (Derived from Power Rule): $\log_b(\sqrt[n]{x}) = \log_b(x^{\frac{1}{n}}) = \frac{1}{n}\log_b(x)$ - The logarithm of a root is the reciprocal of the root's index times the logarithm of the number.

These properties are the cornerstones of expanding logarithmic expressions, and mastering them is crucial for success. Let's get started, guys!

Step-by-Step Expansion of the Logarithmic Expression

Now, let's tackle the expression $\log _c\left(\sqrt[4]{\frac{m^{16} n^{16}}{c^3 a^5}}\right)$ step-by-step. We'll apply the logarithmic properties one at a time to break it down.

Step 1: Apply the Root Rule

First, we have a fourth root, which can be rewritten as a fractional exponent. Applying the root rule (or the power rule, considering the root as a fractional power), we get:

$\log _c\left(\sqrt[4]{\frac{m^{16} n^{16}}{c^3 a^5}}\right) = \log _c\left(\left(\frac{m^{16} n^{16}}{c^3 a^5}\right)^{\frac{1}{4}}\right) = \frac{1}{4} \log _c\left(\frac{m^{16} n^{16}}{c^3 a^5}\right)$

We've successfully moved the exponent outside the logarithm, simplifying the expression inside.

Step 2: Apply the Quotient Rule

Next, we have a fraction inside the logarithm. We can use the quotient rule to separate the numerator and the denominator:

$\frac{1}{4} \log _c\left(\frac{m^{16} n^{16}}{c^3 a^5}\right) = \frac{1}{4} \left[\log _c(m^{16} n^{16}) - \log _c(c^3 a^5)\right]$

Remember to keep the $\frac{1}{4}$ outside the entire expression, as it applies to both terms resulting from the quotient rule.

Step 3: Apply the Product Rule (Twice)

Now we have products in both logarithms. Let's apply the product rule to expand them further:

$\frac{1}{4} \left[\log _c(m^{16} n^{16}) - \log _c(c^3 a^5)\right] = \frac{1}{4} \left[\log _c(m^{16}) + \log _c(n^{16}) - (\log _c(c^3) + \log _c(a^5))\right]$

It's crucial to keep the parentheses around the second group of logarithms because the subtraction applies to the entire group.

Step 4: Apply the Power Rule (Multiple Times)

We're almost there! Now we have logarithms of terms raised to powers. Applying the power rule to each term, we get:

$\frac{1}{4} \left[\log _c(m^{16}) + \log _c(n^{16}) - (\log _c(c^3) + \log _c(a^5))\right] = \frac{1}{4} \left[16 \log _c(m) + 16 \log _c(n) - (3 \log _c(c) + 5 \log _c(a))\right]$

Step 5: Simplify and Distribute

Remember that $\log_c(c) = 1$ . We can substitute that in and then distribute the $\frac{1}{4}$ :

$\frac{1}{4} \left[16 \log _c(m) + 16 \log _c(n) - (3 \log _c(c) + 5 \log _c(a))\right] = \frac{1}{4} \left[16 \log _c(m) + 16 \log _c(n) - (3(1) + 5 \log _c(a))\right]$

$\frac{1}{4} \left[16 \log _c(m) + 16 \log _c(n) - 3 - 5 \log _c(a)\right] = 4 \log _c(m) + 4 \log _c(n) - \frac{3}{4} - \frac{5}{4} \log _c(a)$

The Final Expanded Form

Therefore, the expanded form of the given logarithmic expression is:

$4 \log _c(m) + 4 \log _c(n) - \frac{3}{4} - \frac{5}{4} \log _c(a)$

Common Mistakes to Avoid

Expanding logarithmic expressions can be tricky, and it's easy to make mistakes. Here are some common pitfalls to watch out for:

Forgetting to Distribute: When a constant is multiplied by a group of logarithms (as in Step 5), remember to distribute the constant to every term inside the parentheses.
Incorrectly Applying the Quotient Rule: Ensure you are subtracting the logarithm of the entire denominator, not just the first term.
Mixing Up Product and Quotient Rules: Remember that the logarithm of a product becomes a sum, and the logarithm of a quotient becomes a difference.
Incorrectly Simplifying $\bf{\log_b(b)}$ : Remember, the logarithm of a base to itself is always 1 (e.g., $\log_c(c) = 1$ ).

By avoiding these mistakes, you can confidently expand logarithmic expressions.

Practice Problems

To solidify your understanding, try expanding the following logarithmic expressions:

$\log_b(\frac{x^3 y^2}{\sqrt{z}})$
$\ln(\sqrt[5]{a^2 b^3 c})$
$\log(\frac{100x^4}{y^2 z})$

Work through these problems step-by-step, applying the properties of logarithms we discussed. Check your answers with a calculator or online resources to ensure you've got it right.

Conclusion

Expanding logarithmic expressions is a fundamental skill in mathematics. By understanding and applying the product, quotient, power, and root rules, you can break down complex logarithms into simpler, more manageable terms. Remember to pay close attention to detail and avoid common mistakes. With practice, you'll become a pro at expanding logarithms! This skill is super useful, guys, so keep practicing! You've got this!

This detailed guide should provide a comprehensive understanding of how to express the given logarithmic expression in terms of sums and differences of logarithms. Remember to practice regularly to master these skills. You'll be rocking those logarithmic equations in no time!

In summary, by consistently applying the properties of logarithms – the product rule, the quotient rule, the power rule, and remembering the crucial simplification that $\log_b(b) = 1$ – you can confidently transform complex logarithmic expressions into sums and differences of simpler terms. This ability is not just a mathematical exercise; it’s a powerful tool for solving equations, simplifying models, and understanding the relationships between variables in various scientific and engineering contexts. The journey to mastering logarithms is one of consistent application and a keen eye for detail. So, embrace the challenge, work through those practice problems, and watch your mathematical skills grow!