Problem 27 - Precalculus

Question

Use the properties of logarithms to write the expression as a single logarithm. $$ \log _{2}(x)+\log _{4}(x) $$

Step-by-Step Solution

Answer

The expression is \( \log_2(x^{3/2}) \).

Step 1: Convert Base of Logarithms

To combine the logarithms, they need to have the same base. Use the change of base formula to convert \( \log_4(x) \) to base 2. Recall that the change of base formula is \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \). So, \( \log_4(x) = \frac{\log_2(x)}{\log_2(4)} \). Since \( 4 = 2^2 \), \( \log_2(4) = 2 \). Therefore, \( \log_4(x) = \frac{\log_2(x)}{2} \).

Step 2: Combine the Logarithms

Now that both logarithms have the same base, \( \log_2 \), they can be combined using the properties of logarithms. Specifically, \( \log_b(a) + \log_b(c) = \log_b(a \times c) \). So, we have \[ \log_2(x) + \frac{\log_2(x)}{2} = \frac{2\log_2(x)}{2} + \frac{\log_2(x)}{2} = \frac{3\log_2(x)}{2}. \]

Step 3: Express as a Single Logarithm

The expression \( \frac{3\log_2(x)}{2} \) can be written as a single logarithm using the power rule of logarithms: \( a\log_b(c) = \log_b(c^a) \). Thus, \( \frac{3\log_2(x)}{2} = \log_2(x^{3/2}) \).

Key Concepts

Change of Base Formula

The change of base formula is a handy tool that allows you to convert a logarithm from one base to another. Imagine you're trying to read a recipe in a foreign language; the change of base formula is like a translation that helps you understand. The formula is given by:
\[\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\]Here, you can change the logarithm of a number with base \(b\) to any new base \(c\). In the original exercise, we used this formula to convert \( \log_4(x)\) into a logarithm with base 2.

• First, we needed \( \log_2(4) \). We noticed that 4 is the same as \( 2^2 \), so \( \log_2(4) = 2 \).
• The change of base formula helped us rewrite \( \log_4(x) \) as \( \frac{\log_2(x)}{2} \).

This conversion was crucial for combining the logarithms in the next step. By ensuring the bases were the same, we simplified our task enormously.

Combining Logarithms

Once you have logarithms with identical bases, you can combine them much more easily. In math, this is like having all your ingredients ready before cooking a meal—it makes the process smoother. The property used is:
\[ \log_b(a) + \log_b(c) = \log_b(a \times c) \]In our example, after converting \( \log_4(x) \) to base 2, we had:

• \( \log_2(x) + \frac{\log_2(x)}{2} \)

By expressing them over a common denominator, we combined them as:
\[\frac{2\log_2(x)}{2} + \frac{\log_2(x)}{2} = \frac{3\log_2(x)}{2}\]This combined expression re-arranges and simplifies the elements into a single entity, setting the stage for our final transformation.

Power Rule of Logarithms

The power rule of logarithms is like a magical transformation that allows us to write multiplied logarithms as just one beautiful expression. It says:
\[ a\log_b(c) = \log_b(c^a) \]This rule enables us to bring any coefficients or factors within the logarithm itself. In our case, we had:
\( \frac{3\log_2(x)}{2} \)

• The power rule lets us move the \(\frac{3}{2}\) out of the front, and into the power position: \( \log_2(x^{3/2}) \).

This effectively allowed us to represent the entire expression as a single logarithm. Such simplification is beneficial when solving complex equations, as it reduces clutter and emphasizes clarity, making your solutions more elegant and easy to follow.