Problem 27 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

\( \log_{5}(x^3 z^6) \)

Step 1: Apply the Power Rule of Logarithms

The power rule for logarithms states that for a logarithm with a coefficient, the coefficient can be used as an exponent on the argument of the logarithm. Rewrite the given expression by applying the power rule:\[ 3 \log_{5} x = \log_{5} (x^3) \]\[ 6 \log_{5} z = \log_{5} (z^6) \]

Step 2: Apply the Product Rule of Logarithms

The product rule for logarithms tells us that the sum of two logarithms with the same base can be rewritten as a single logarithm of the product of their arguments: \[ \log_{5} (x^3) + \log_{5} (z^6) = \log_{5} (x^3 \times z^6) \]

Step 3: Combine into a Single Expression

Combine the entire expression into a single logarithmic expression:\[ \log_{5} (x^3 z^6) \]

Key Concepts

Power Rule of Logarithms

When working with logarithmic expressions, a common occurrence is having a coefficient in front of the logarithm. The power rule of logarithms is a helpful tool that allows us to simplify such expressions. This rule states:

• If you have a logarithm of the form \( a \log_b M \), it can be rewritten as \( \log_b (M^a) \).

This means the coefficient becomes an exponent on the argument of the logarithm.
For example, in the exercise \( 3 \log_5 x \), we use the power rule to transform it into \( \log_5 (x^3) \).
This makes the expression easier to work with, especially when combining logarithms. Remember this handy rule whenever you encounter coefficients in your logarithmic expressions.

Product Rule of Logarithms

The product rule of logarithms is a nifty tool when you need to combine two logarithms with the same base. This rule states:

• If you have \( \log_b M + \log_b N \), you can combine them into a single logarithm: \( \log_b (M \cdot N) \).

This transformation is particularly useful for simplifying expressions with multiple logarithms.
In our step-by-step solution, we applied this rule to combine \( \log_5 (x^3) \) and \( \log_5 (z^6) \).
By applying the product rule, these two logarithms became a single expression: \( \log_5 (x^3 \cdot z^6) \). Ensure that both logarithms initially have the same base before applying the product rule. It's a useful shortcut for simplifying your work.

Combining Logarithms

Combining logarithms is an essential skill in algebra, specifically when dealing with large and complex expressions. When you have logarithms with the same base, you can often merge them into a single logarithm expression using a combination of rules (power and product). Here is how it works:

• First, if needed, use the power rule to simplify any coefficients.
• Second, use the product rule to merge the simplified logarithms into one.

In our example, we started with \( 3 \log_5 x + 6 \log_5 z \). By applying both the power and product rules, we reduced it to \( \log_5 (x^3z^6) \).
This process not only makes the expression compact but also easier to handle in further calculations.
Practice and familiarity with these rules will make tackling logarithmic expressions a breeze.