Problem 37 - Algebra A Combined Function

Question

Write each as a single logarithm. Assume that variables represent positive numbers. $$ 2 \log _{8} x-\frac{2}{3} \log _{8} x+4 \log _{8} x $$

Step-by-Step Solution

Answer

\( \log_{8}(x^{\frac{16}{3}}) \)

Step 1: Factor the Common Logarithm

Each term in the expression \(2\log_{8}x - \frac{2}{3}\log_{8}x + 4\log_{8}x\) contains the common logarithmic term \(\log_{8}x\). We can factor this out. The expression becomes:\[(2 - \frac{2}{3} + 4)\log_{8}x\]

Step 2: Simplify the Coefficients

Simplify the coefficients within the parentheses: \[2 - \frac{2}{3} + 4 = \frac{6}{3} - \frac{2}{3} + \frac{12}{3}\]Combine these to find a common denominator:\[= \frac{6}{3} - \frac{2}{3} + \frac{12}{3} = \frac{16}{3}\]

Step 3: Rewrite as a Single Logarithm

Using the simplified coefficient, rewrite the original expression as a single logarithm:\[ (\frac{16}{3})\log_{8}x = \log_{8}x^{\frac{16}{3}} \]

Step 4: Final Expression as a Single Logarithm

The entire expression \(2\log_{8}x - \frac{2}{3}\log_{8}x + 4\log_{8}x\) is now simplified to a single logarithm:\[ \log_{8}(x^{\frac{16}{3}}) \]

Key Concepts

Factor the Common Term

When dealing with expressions involving logarithms, a useful step can be to factor out common terms. In the original expression

• \(2\log_{8}x - \frac{2}{3}\log_{8}x + 4\log_{8}x\)

we notice that each term contains the common logarithmic term \(\log_{8}x\). This means that \(\log_{8}x\) appears in every piece of the expression.
By factoring it out, we streamline the expression, making it easier to deal with. Performing this factoring step, the expression converts to:

• \((2 - \frac{2}{3} + 4)\log_{8}x\)

Now, we have a single logarithmic term multiplied by a set of coefficients inside the parentheses. It simplifies both the visual clutter and your subsequent algebraic manipulations.

Simplify Coefficients

After factoring out the common term, the next step involves simplifying the coefficients obtained inside the parentheses. In the expression:

• \((2 - \frac{2}{3} + 4)\log_{8}x\)

the coefficients to be simplified are \(2\), \(-\frac{2}{3}\), and \(4\).
Combining these necessitates finding a common denominator. This is crucial for simplifying the coefficients properly. For the numbers \(2\), \(-\frac{2}{3}\), and \(4\), the common denominator is \(3\). Rewriting the coefficients gives us:

• \(\frac{6}{3} - \frac{2}{3} + \frac{12}{3}\)

Performing the addition and subtraction results in \(\frac{16}{3}\).
Therefore, the expression is simplified correctly, paving the way for the final simplification into a single logarithm.

Expressions as Single Logarithm

The last step in the process is rewriting the expression as a single logarithm. With the simplified coefficient \(\frac{16}{3}\) at hand, we can consolidate the expression:

• \((\frac{16}{3})\log_{8}x\)

The property of logarithms used here is that multiplying a logarithm by a coefficient is equivalent to raising the argument inside the logarithm to the power of that coefficient. This is a powerful property of logarithms that allows us to rewrite the expression:

• \(\log_{8}x^{\frac{16}{3}}\)

Thus, the expression originally given \(2\log_{8}x - \frac{2}{3}\log_{8}x + 4\log_{8}x\) is elegantly simplified into a single logarithm showing how logarithmic rules and properties can be effectively applied to simplify expressions.