Problem 29 - Algebra A Combined Function

Question

Write each as a single logarithm. Assume that variables represent positive numbers. $$ \log _{4} 2+\log _{4} 10-\log _{4} 5 $$

Step-by-Step Solution

Answer

The expression simplifies to \( \log_4 4 = 1 \).

Step 1: Use the Logarithm Addition Rule

The log addition rule states \( \log_b A + \log_b B = \log_b (A \times B) \). Apply this to the \( \log_4 2 + \log_4 10 \) to get \( \log_4 (2 \times 10) \).

Step 2: Simplify the Product in the Logarithm

Compute the product inside the logarithm: \( 2 \times 10 = 20 \). So the expression becomes \( \log_4 20 \).

Step 3: Apply the Logarithm Subtraction Rule

The log subtraction rule states \( \log_b A - \log_b B = \log_b (\frac{A}{B}) \). Use this on \( \log_4 20 - \log_4 5 \) to obtain \( \log_4 (\frac{20}{5}) \).

Step 4: Simplify the Division Inside the Logarithm

Calculate the division inside the logarithm: \( \frac{20}{5} = 4 \). Thus, simplify the expression to \( \log_4 4 \).

Step 5: Evaluate the Simplified Logarithm

Recognize that \( \log_4 4 = 1 \), because 4 is the base and \( 4^1 = 4 \).

Key Concepts

Logarithm Addition Rule

When working with logarithms, understanding the addition rule is pivotal. The logarithm addition rule states that for the same base, you can add two logarithms by multiplying their arguments and keeping the same base for the new logarithm.
The rule looks like this:

• \( \log_b (A) + \log_b (B) = \log_b (A \times B) \)

If you have \( \log_4 2 \) and \( \log_4 10 \), combining them using this rule will result in \( \log_4 (2 \times 10) \).
You multiply 2 and 10 to get 20, leading to \( \log_4 20 \).
This rule stems from the fact that logarithms essentially count the powers needed for a base to reach an argument's value.
Thus, multiplying the arguments corresponds to adding their component powers.

Logarithm Subtraction Rule

Much like the addition rule, the subtraction rule for logarithms helps to process expressions effectively. This rule indicates that subtracting two logarithms with the same base is equivalent to dividing their arguments inside a single logarithm.
The rule can be expressed as:

• \( \log_b (A) - \log_b (B) = \log_b \left( \frac{A}{B} \right) \)

In the exercise, we've arrived at \( \log_4 20 \) and need to subtract \( \log_4 5 \).
According to the rule, this simplifies to \( \log_4 \left( \frac{20}{5} \right) \).
By calculating \( \frac{20}{5} \), we get 4. Thus, the expression further simplifies to \( \log_4 4 \).
Understanding this concept is crucial as it helps in simplifying complex logarithmic expressions quickly by recognizing them in these common formats.

Simplifying Logarithms

After applying the addition and subtraction rules, the final critical step is to simplify the resulting logarithm.
Simplification of logarithms means reducing them to their simplest form so that they are easily manageable or recognizable.

• The key simplification at the end of the exercise involves recognizing \( \log_4 4 \).
• Considering the base 4, \( \log_4 4 \) asks how many times you need to multiply 4 by itself to get 4 again. It’s just once, so \( \log_4 4 = 1 \).

This simplification step is crucial, especially in solving problems or proving identities in logarithms, as it brings the expression to an easy to interpret and complete conclusion.
A strong grasp of simplifying logarithms improves speed and accuracy in mathematical problem solving, enhancing overall numerical literacy.