Problem 8 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

\( \log_{6}(15x + 60) \)

Step 1: Apply the Product Rule for Logarithms

The product rule for logarithms states that the sum of logarithms with the same base can be expressed as a single logarithm of the product of their arguments. So, for the given expression, use this rule: \[\log_{6} 3 + \log_{6} (x+4) + \log_{6} 5 = \log_{6}(3 \cdot (x+4) \cdot 5)\]

Step 2: Calculate the Product Inside the Logarithm

Simplify the expression inside the logarithm by multiplying the numbers and variable expression:\[3 \cdot (x+4) \cdot 5 = 15(x+4)\]This simplifies to:\[15x + 60\]

Step 3: Write the Expression as a Single Logarithm

Now, substitute the simplified expression from Step 2 back into the logarithmic expression:\[\log_{6}(15x + 60)\]

Key Concepts

Product Rule for Logarithms

When dealing with logarithms, the product rule is a helpful tool to simplify expressions. The product rule states that the sum of logarithms with the same base can be rewritten as a single logarithm of the product of their arguments.
This means if you have an expression like \( \log_a b + \log_a c \), it can be combined to \( \log_a(b \cdot c) \).

For the original exercise, we had three terms: \( \log_6 3 + \log_6 (x+4) + \log_6 5 \). Since they all share the base 6, we can combine them into one logarithm by multiplying all their arguments:
\[ \log_6 (3 \cdot (x+4) \cdot 5) \]
This step is crucial in reducing the complexity of logarithmic expressions and preparing them for further simplification.

Sum of Logarithms

The concept of the sum of logarithms extends the product rule to more than just two terms.
When you encounter a sum of multiple logarithms with the same base, you can apply the product rule repeatedly.

This is useful because it allows you to transform a lengthy expression into a compact form. In our exercise, we had three logarithmic terms added together.
By using the product rule, we expressed this sum as a single logarithm with the base 6:
\[ \log_6(3 \cdot (x+4) \cdot 5) \]
This product combines both numerical and variable components into one clean expression.
Seeing logarithms in their summed form helps to understand better how they interact and can be manipulated.

Simplifying Logarithmic Expressions

After combining the logarithms into a single expression, the next step is often simplification.
Simplifying involves calculating the product inside the logarithm by multiplying all the numbers and expressions together.

In our case, the expression inside the logarithm was:
\[ 3 \cdot (x+4) \cdot 5 = 15(x+4) \]
Further simplification leads to:
\[ 15x + 60 \]
This step shows the importance of fully expanded products, making it easier to understand or calculate further.
Always remember the goal is to present the expression in its simplest, most readable form, which reduces errors in calculations going forward.
By expressing \( \log_6(15x + 60) \), we achieve a streamlined view of the relationship between the components of the problem.