Rewriting Logarithmic Expressions As A Single Logarithm

#mainkeyword Logarithmic expressions are fundamental in mathematics, particularly when dealing with exponential functions and equations. The properties of logarithms provide powerful tools for simplifying and manipulating these expressions. This article delves into how to use these properties to rewrite a given expression as a single logarithm and evaluate logarithmic expressions whenever possible. We will focus on the expression $\ln \sqrt{x} - \frac{1}{4} \ln x + \ln \sqrt[6]{x}$, breaking down each step to ensure clarity and understanding.

Understanding the Properties of Logarithms

Before we dive into the specific expression, it's crucial to understand the core properties of logarithms. These properties allow us to combine, expand, and simplify logarithmic expressions, which is essential for solving equations and simplifying complex mathematical models. Let's explore the key properties that we will use:

1. Product Rule: This rule states that the logarithm of the product of two numbers is equal to the sum of their logarithms. Mathematically, it is expressed as $\log_b(MN) = \log_b(M) + \log_b(N)$. This property is particularly useful for expanding a single logarithm into multiple logarithms.
2. Quotient Rule: Conversely, the logarithm of the quotient of two numbers is equal to the difference of their logarithms. The mathematical representation is $\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$. This rule helps in breaking down a logarithm of a fraction into simpler terms.
3. Power Rule: The power rule states that the logarithm of a number raised to an exponent is equal to the exponent multiplied by the logarithm of the number. This is written as $\log_b(M^p) = p \log_b(M)$. The power rule is incredibly useful for simplifying expressions with exponents inside logarithms.
4. Change of Base Rule: Although not directly used in this specific problem, it is worth mentioning. This rule allows us to change the base of a logarithm, which is especially useful when dealing with calculators that only compute common logarithms (base 10) or natural logarithms (base e). The rule is $\log_b(M) = \frac{\log_k(M)}{\log_k(b)}$, where k is the new base.
5. Logarithm of 1: For any base b, $\log_b(1) = 0$. This is because any number raised to the power of 0 is 1.
6. Logarithm of the Base: For any base b, $\log_b(b) = 1$. This is because any number raised to the power of 1 is itself.

These properties are the foundation for manipulating logarithmic expressions. Mastering them is essential for simplifying complex expressions and solving logarithmic equations. In the following sections, we will apply these properties to rewrite the given expression as a single logarithm.

Rewriting the Expression: $\ln \sqrt{x} - \frac{1}{4} \ln x + \ln \sqrt[6]{x}$

Let's apply these properties to rewrite the expression $\ln \sqrt{x} - \frac{1}{4} \ln x + \ln \sqrt[6]{x}$ as a single logarithm. Our goal is to combine the three logarithmic terms into one, making the expression simpler and easier to work with. We'll proceed step-by-step, using the power rule, followed by the product and quotient rules.

Step 1: Applying the Power Rule

The first step is to use the power rule to address the exponents within the logarithms. Recall that the power rule states $\log_b(M^p) = p \log_b(M)$. We have square roots and a fractional exponent in our expression. We'll rewrite the roots as fractional exponents to make the application of the power rule clearer:

• $\sqrt{x}$ is equivalent to $x^{\frac{1}{2}}$
• $\sqrt[6]{x}$ is equivalent to $x^{\frac{1}{6}}$

Now we can rewrite the original expression using these fractional exponents:

$\ln (x^{\frac{1}{2}}) - \frac{1}{4} \ln x + \ln (x^{\frac{1}{6}})$

Next, we apply the power rule to each term:

• $\ln (x^{\frac{1}{2}})$ becomes $\frac{1}{2} \ln x$
• $\ln (x^{\frac{1}{6}})$ becomes $\frac{1}{6} \ln x$

Substituting these back into the expression, we get:

$\frac{1}{2} \ln x - \frac{1}{4} \ln x + \frac{1}{6} \ln x$

Step 2: Combining Like Terms

Now that we have applied the power rule, we can see that all terms have a common factor of $\ln x$. This allows us to combine these terms by treating $\ln x$ as a common variable. We need to find a common denominator for the fractions $\frac{1}{2}$, $\frac{1}{4}$, and $\frac{1}{6}$. The least common denominator (LCD) for 2, 4, and 6 is 12. So, we rewrite the fractions with the common denominator:

• $\frac{1}{2} = \frac{6}{12}$
• $\frac{1}{4} = \frac{3}{12}$
• $\frac{1}{6} = \frac{2}{12}$

Now our expression looks like this:

$\frac{6}{12} \ln x - \frac{3}{12} \ln x + \frac{2}{12} \ln x$

Combine the fractions:

$(\frac{6}{12} - \frac{3}{12} + \frac{2}{12}) \ln x$

$(\frac{6 - 3 + 2}{12}) \ln x$

$\frac{5}{12} \ln x$

Step 3: Applying the Power Rule (in Reverse)

We have now simplified the expression to a single term with a coefficient: $\frac{5}{12} \ln x$. To rewrite this as a single logarithm, we use the power rule in reverse. This means we take the coefficient and make it the exponent of the argument inside the logarithm. Recall the power rule: $p \log_b(M) = \log_b(M^p)$.

Applying this to our expression, we get:

$\ln (x^{\frac{5}{12}})$

This is now a single logarithm. We can further rewrite the fractional exponent as a root if desired:

$x^{\frac{5}{12}}$ is equivalent to $\sqrt[12]{x^5}$

So, our final rewritten expression is:

$\ln (\sqrt[12]{x^5})$

Final Answer

Therefore, the expression $\ln \sqrt{x} - \frac{1}{4} \ln x + \ln \sqrt[6]{x}$ can be rewritten as a single logarithm: $\ln (\sqrt[12]{x^5})$. This demonstrates the power of using logarithmic properties to simplify and consolidate expressions. By applying the power rule, combining like terms, and understanding the relationship between exponents and logarithms, we were able to transform a complex expression into a single, manageable term.

Conclusion

In summary, using the properties of logarithms such as the power rule, product rule, and quotient rule, allows us to rewrite and simplify complex expressions into single logarithmic terms. In the given example, $\ln \sqrt{x} - \frac{1}{4} \ln x + \ln \sqrt[6]{x}$, we successfully consolidated the expression into a single logarithm: $\ln (\sqrt[12]{x^5})$. This process involves converting roots to fractional exponents, applying the power rule, finding common denominators to combine terms, and then rewriting the expression using the power rule in reverse. Mastering these techniques is essential for anyone working with logarithmic and exponential functions. The ability to manipulate logarithmic expressions not only simplifies calculations but also provides deeper insights into the relationships between variables in mathematical models and real-world applications. Understanding these logarithmic properties is crucial for further studies in mathematics, physics, engineering, and other quantitative fields.

By following these steps, you can confidently tackle similar problems and gain a deeper understanding of logarithmic functions and their applications. Remember, the key is to practice and become familiar with the properties, which will make the simplification process more intuitive and efficient.