Problem 35 - Algebra A Combined Function

Question

Write each as a single logarithm. Assume that variables represent positive numbers. $$ 3 \log _{2} x+\frac{1}{2} \log _{2} x-2 \log _{2}(x+1) $$

Step-by-Step Solution

Answer

The expression simplifies to \(\log_{2} \frac{x^{7/2}}{(x+1)^2}\).

Step 1: Identify the expression

The given expression is \(3 \log_{2} x + \frac{1}{2} \log_{2} x - 2 \log_{2}(x+1)\). Our goal is to write this as a single logarithm.

Step 2: Apply the Power Rule

The power rule of logarithms states that \(a \log_{b} c = \log_{b} c^{a}\). Apply this, we have:\[ 3 \log_{2} x = \log_{2} x^{3}, \quad \frac{1}{2} \log_{2} x = \log_{2} x^{1/2}, \quad 2 \log_{2} (x+1) = \log_{2} (x+1)^2 \]

Step 3: Combine the Logarithms Using the Product Rule

Combine \(\log_{2} x^{3}\) and \(\log_{2} x^{1/2}\) using the product rule, which states \(\log_b a + \log_b c = \log_b (ac)\). As a result, we have:\[ \log_{2} x^{3} + \log_{2} x^{1/2} = \log_{2} (x^{3} \cdot x^{1/2}) = \log_{2} x^{7/2} \]

Step 4: Apply the Quotient Rule

Now utilize the quotient rule, which states that \(\log_b a - \log_b c = \log_b \left(\frac{a}{c}\right)\). Substract the third term from the result of Step 3:\[ \log_{2} x^{7/2} - \log_{2} (x+1)^2 = \log_{2} \frac{x^{7/2}}{(x+1)^2} \]

Step 5: Write the Final Combined Expression

The expression is now combined into a single logarithm:\[ \log_{2} \frac{x^{7/2}}{(x+1)^2} \]

Key Concepts

Power Rule

The power rule is a fundamental aspect of logarithms that simplifies expressions that contain multiplied constants. It essentially allows you to "move" a coefficient to an exponent position. For example, in the expression \( a \log_{b} c \), using the power rule, you can rewrite it as \( \log_{b} c^a \). This is very useful when you encounter logarithms that have coefficients in front of them.In our original exercise, we applied the power rule to each term:

• \( 3 \log_{2} x \) becomes \( \log_{2} x^{3} \)
• \( \frac{1}{2} \log_{2} x \) becomes \( \log_{2} x^{1/2} \)
• \( 2 \log_{2} (x+1) \) transforms into \( \log_{2} (x+1)^{2} \)

These transformations make it easier to handle and simplify the logarithmic expression further.

Product Rule

Once we've used the power rule, the next logical step is often the product rule. This useful property of logarithms is employed to combine multiple logarithmic expressions into one. The product rule states that \( \log_b a + \log_b c = \log_b(ac) \). This allows you to add two logs together into a singular log expression, representing the product of the two original values.In the exercise, it was used to combine:

• \( \log_{2} x^{3} \)
• \( \log_{2} x^{1/2} \)

This combines to give us \( \log_{2} (x^{3} \cdot x^{1/2}) = \log_{2} x^{7/2} \). Whenever you see an addition of logs with the same base, think of combining using the product rule for simplification.

Quotient Rule

Finally, the quotient rule is the complement to the product rule and equally important when simplifying logarithmic expressions. This rule applies when you need to subtract one logarithm from another with the same base. It is written as \( \log_b a - \log_b c = \log_b \left(\frac{a}{c}\right) \).In our solution, after simplifying the first two terms with the power and product rules, the expression was further simplified using the quotient rule:

• The previously combined term \( \log_{2} x^{7/2} \)
• Subtracting the term \( \log_{2} (x+1)^2 \)

This results in \( \log_{2} \frac{x^{7/2}}{(x+1)^2} \). This process of subtraction using the quotient rule helps reduce complex multi-logarithm expressions into a single, more manageable form.