Problem 15 - Precalculus Functions and Graphs

Question

Rewrite each expression as a single logarithm. $$\log _{4}(12)-\log _{4}(2)$$

Step-by-Step Solution

Answer

\(\log_4 (6)\).

Step 1 - Recall the Logarithm Quotient Rule

The logarithm quotient rule states that \(\log_b (\frac{M}{N}) = \log_b M - \log_b N\). This rule can be used to combine the two logarithms into one.

Step 2 - Apply the Quotient Rule

Using the quotient rule, \(\log_4 (12) - \log_4 (2)\) can be rewritten as \(\log_4 (\frac{12}{2}) = \log_4 6\).

Key Concepts

Logarithm Quotient Rule

To understand the logarithm quotient rule, it's essential to know that logarithms are the inverse of exponentiation. The quotient rule helps us simplify the difference of two logarithms into a single logarithm.
The rule is expressed as: \[\text{log}_b \frac{M}{N} = \text{log}_b M - \text{log}_b N\]
Here, \(\text{log}_b M\) and \(\text{log}_b N\) are two logarithmic expressions with the same base \(b\). According to the rule, subtracting one logarithm from another with the same base equals the logarithm of the quotient of their arguments.
For example: \[\text{log}_4 12 - \text{log}_4 2 = \text{log}_4 \frac{12}{2} = \text{log}_4 6\]
By using this rule, we can simplify seemingly complex expressions into more manageable forms.

Single Logarithm Expression

A single logarithm expression is one that combines multiple logarithms into one. This is useful when simplifying equations or expressions involving logarithms.
Combining logarithms is made possible through logarithm rules, like the product, quotient, and power rules. In our problem: \[\text{log}_4 12 - \text{log}_4 2 \] we use the logarithm quotient rule to combine these into a single logarithm.
Here's how it works step-by-step:

• Use the logarithm quotient rule: \(\text{log}_4 \frac{12}{2}\)
• Simplify the fraction: \(\frac{12}{2} = 6\)
• Result: \(\text{log}_4 6\)

Now, the expression \(\text{log}_4 12 - \text{log}_4 2\) is simplified to the single logarithm \(\text{log}_4 6\) . This makes it easier to work with in further calculations or in solving equations.

Combining Logarithms

Combining logarithms means using logarithmic rules to merge different logarithmic terms into one. This is particularly useful when simplifying expressions or solving equations.
Several rules allow us to combine logarithms:

• The product rule: \[\text{log}_b (MN) = \text{log}_b M + \text{log}_b N\]
• The quotient rule: \[\text{log}_b \frac{M}{N} = \text{log}_b M - \text{log}_b N\]
• The power rule: \[\text{log}_b (M^p) = p \text{log}_b M\]

In our example: \[\text{log}_4 12 - \text{log}_4 2\] we specifically use the quotient rule to combine the two logarithms into a single expression.
Applying the quotient rule:

• Combine terms: \(\text{log}_4 \frac{12}{2}\)
• Simplify: \(\frac{12}{2} = 6\)
• Result: \(\text{log}_4 6\)

This process demonstrates how logarithmic rules like the quotient rule can simplify complex expressions into a more streamlined form.