Express Logarithmic Expressions Without Exponents

Jul 14, 2025 by ADMIN 50 views

Expressing Radicals and Exponents in Logarithmic Form

Let's delve into the process of expressing a logarithmic expression involving radicals and exponents in a simplified form, devoid of exponents. This involves leveraging the fundamental properties of logarithms to break down the expression into manageable components. In this comprehensive guide, we will meticulously dissect the given expression and apply the necessary logarithmic rules to achieve the desired form.

Understanding Logarithmic Properties

Before we embark on the simplification journey, it's crucial to have a firm grasp of the core logarithmic properties that will serve as our tools. These properties are the bedrock of logarithmic manipulation and will enable us to transform the expression effectively.

Product Rule: The logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, this is represented as:

$\log_b(MN) = \log_b(M) + \log_b(N)$
Quotient Rule: The logarithm of a quotient is equal to the difference between the logarithms of the numerator and the denominator. In mathematical notation:

$\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$
Power Rule: The logarithm of a quantity raised to a power is equal to the product of the power and the logarithm of the quantity. Expressed mathematically:

$\log_b(M^p) = p \log_b(M)$
Root Rule: The logarithm of the nth root of a quantity is equal to the logarithm of the quantity divided by n. This can be written as:

$\log_b(\sqrt[n]{M}) = \frac{1}{n} \log_b(M)$

These properties are the key to unlocking the simplification of our target expression. By strategically applying them, we can systematically eliminate exponents and radicals, ultimately expressing the logarithm in its most basic form.

Breaking Down the Expression

Now, let's focus on the given expression:

$\log _c\left(\sqrt[4]{\frac{m^{16} n^8}{c^5}}\right)$

Our mission is to express this in terms of logarithms without any exponents. We'll proceed step-by-step, employing the logarithmic properties we've discussed.

Step 1: Eliminate the Radical

The outermost element of our expression is the fourth root. We can utilize the root rule to rewrite this as a fractional exponent:

$\log _c\left(\sqrt[4]{\frac{m^{16} n^8}{c^5}}\right) = \log _c\left(\left(\frac{m^{16} n^8}{c^5}\right)^{\frac{1}{4}}\right)$

This step replaces the radical with its equivalent exponential form, setting the stage for the next phase of simplification.

Step 2: Apply the Power Rule

Now that we have a fractional exponent, we can leverage the power rule to bring the exponent outside the logarithm:

$\log _c\left(\left(\frac{m^{16} n^8}{c^5}\right)^{\frac{1}{4}}\right) = \frac{1}{4} \log _c\left(\frac{m^{16} n^8}{c^5}\right)$

The power rule effectively moves the exponent from within the logarithm to a coefficient outside, simplifying the expression further.

Step 3: Employ the Quotient Rule

Next, we encounter a fraction within the logarithm. The quotient rule allows us to separate the numerator and denominator:

$\frac{1}{4} \log _c\left(\frac{m^{16} n^8}{c^5}\right) = \frac{1}{4} \left[\log _c(m^{16} n^8) - \log _c(c^5)\right]$

By applying the quotient rule, we transform the logarithm of a fraction into a difference of logarithms, making the expression more amenable to simplification.

Step 4: Utilize the Product Rule

In the first logarithmic term, we have a product of two variables raised to powers. We can apply the product rule to separate these:

$\frac{1}{4} \left[\log _c(m^{16} n^8) - \log _c(c^5)\right] = \frac{1}{4} \left[\log _c(m^{16}) + \log _c(n^8) - \log _c(c^5)\right]$

The product rule decomposes the logarithm of a product into a sum of logarithms, further isolating the variables.

Step 5: Apply the Power Rule Again

We now have individual logarithmic terms with exponents. We can once again use the power rule to move the exponents outside the logarithms:

$\frac{1}{4} \left[\log _c(m^{16}) + \log _c(n^8) - \log _c(c^5)\right] = \frac{1}{4} \left[16 \log _c(m) + 8 \log _c(n) - 5 \log _c(c)\right]$

This application of the power rule eliminates the exponents, bringing us closer to our final simplified form.

Step 6: Simplify the Logarithmic Term

We have a term $\log _c(c)$ , which is a special case. The logarithm of a number to the same base is always 1:

$\log _c(c) = 1$

Substituting this into our expression:

$\frac{1}{4} \left[16 \log _c(m) + 8 \log _c(n) - 5 \log _c(c)\right] = \frac{1}{4} \left[16 \log _c(m) + 8 \log _c(n) - 5(1)\right]$

This simplification streamlines the expression by removing the logarithmic term with a base and argument.

Step 7: Distribute and Finalize

Finally, we distribute the $\frac{1}{4}$ across the terms within the brackets:

$\frac{1}{4} \left[16 \log _c(m) + 8 \log _c(n) - 5\right] = 4 \log _c(m) + 2 \log _c(n) - \frac{5}{4}$

We have now successfully expressed the original logarithmic expression in terms of logarithms without exponents. This is our simplified answer.

Final Answer

Therefore, the expression $\log _c\left(\sqrt[4]{\frac{m^{16} n^8}{c^5}}\right)$ can be expressed in terms of logarithms without exponents as:

$\boxed{4 \log _c(m) + 2 \log _c(n) - \frac{5}{4}}$

This detailed breakdown demonstrates the power of logarithmic properties in simplifying complex expressions. By systematically applying these rules, we can transform intricate forms into manageable and transparent results. This skill is invaluable in various mathematical and scientific contexts where logarithmic manipulations are frequently encountered.

Conclusion

In conclusion, this article has provided a comprehensive guide on expressing logarithms without exponents, focusing on a specific example. By understanding and applying the fundamental properties of logarithms, such as the product rule, quotient rule, power rule, and root rule, you can effectively simplify complex logarithmic expressions. The step-by-step approach outlined in this article ensures clarity and accuracy in the simplification process. Mastering these techniques is crucial for anyone working with logarithmic functions in mathematics, science, or engineering. Remember, the key to success lies in a solid understanding of the logarithmic properties and their strategic application.