Problem 38 - Algebra A Combined Function

Question

Write each as a single logarithm. Assume that variables represent positive numbers. $$ 5 \log _{6} x-\frac{3}{4} \log _{6} x+3 \log _{6} x $$

Step-by-Step Solution

Answer

\(\log _{6} x^{\frac{29}{4}}\)

Step 1: Write the expression

Identify the expression you need to simplify: \[5 \log _{6} x - \frac{3}{4} \log _{6} x + 3 \log _{6} x\]

Step 2: Combine the coefficients of the logarithms

Notice that all terms involve \( \log _{6} x \). You can factor this out to combine the terms: \[(5 - \frac{3}{4} + 3) \log _{6} x\]

Step 3: Simplify the expression inside the parenthesis

Calculate the simplified coefficient of \( \log _{6} x \) by adding the numbers: \[5 - \frac{3}{4} + 3\] Convert 5 and 3 to fractions with a common denominator (4): \[\frac{20}{4} - \frac{3}{4} + \frac{12}{4} = \frac{29}{4}\]

Step 4: Combine all into a single logarithm

The expression becomes: \[\frac{29}{4} \log _{6} x\] Using the power rule for logarithms, rewrite it as: \[\log _{6} x^{\frac{29}{4}}\]

Step 5: Conclusion

The expression \(5 \log _{6} x - \frac{3}{4} \log _{6} x + 3 \log _{6} x\) simplifies to a single logarithm: \[\log _{6} x^{\frac{29}{4}}\]

Key Concepts

Simplifying Logarithmic Expressions

When dealing with complex logarithmic expressions, it is crucial to simplify them to make calculations more manageable and to better understand the underlying mathematical concepts. Let's take a look at an example to see how this works in practice. Consider the expression:

• \(5 \log_{6} x - \frac{3}{4} \log_{6} x + 3 \log_{6} x\)

In this expression, all the logarithmic terms share the same base \(\log_{6} x\). Our goal is to combine these terms into a single, simplified logarithm by first consolidating the coefficients.

• Identify all terms with the same base.
• Add or subtract the coefficients as needed.

Once the coefficients are combined, you can use properties of logarithms to further simplify the expression, ultimately turning it into a single logarithmic expression, like \(\log_{6} x^{\frac{29}{4}}\).

Logarithm Properties

Logarithms have several fundamental properties that can simplify complex expressions and solve logarithmic equations. The two properties relevant to our example are the power rule and the property of logarithms for addition and subtraction. The **power rule** states that \(a \cdot \log_{b} C = \log_{b} C^{a}\). Essentially, a coefficient in front of a logarithm can be rewritten as an exponent inside the log.

• This property allows you to handle multiplicative factors by converting them into powers, simplifying calculations.

In our exercise, the power rule was used to rewrite the expression \(\frac{29}{4} \log_{6} x\) to a more compact form: \(\log_{6} x^{\frac{29}{4}}\).Another basic property is the **addition/subtraction property**, which allows us to combine the logarithms with the same base by adding or subtracting their coefficients first. This way, the expression can be factored and combined neatly:

• Combine: \(5 - \frac{3}{4} + 3\).
• Simplify to a single term.

These properties make it easier to deal with logarithmic expressions efficiently.

Algebraic Manipulation

Algebraic manipulation is a powerful tool that helps in transforming expressions into more usable forms. In the context of logarithms, algebraic manipulation involves factorizing, combining like terms, and converting expressions just as we do with regular numbers. In our example, manipulation techniques included:

• Factor out the common logarithmic term: \(\log_{6} x\).
• Combine the coefficients \(5\), \(-\frac{3}{4}\), and \(3\) by rewriting them with a common denominator.

This involves combining these coefficients: \((5 - \frac{3}{4} + 3)\) by converting them into fractions to find their sum which results in \(\frac{29}{4}\).Algebraic manipulation also uses the beauty of the common denominator trick in fractions, which ensures accurate and easy addition or subtraction of fraction numbers. With careful manipulation, complex logarithmic expressions are made less intimidating and calculable. By practicing algebraic manipulation, you gain skills to tackle a wide variety of math problems efficiently.