Problem 16 - Algebra and Trigonometry

Question

\(15-24\) Evaluate the expression. $$ \begin{array}{llll}{\text { (a) } \log _{5} 5^{4}} & {\text { (b) } \log _{4} 64} & {\text { (c) } \log _{9} 9}\end{array} $$

Step-by-Step Solution

Answer

(a) 4, (b) 3, (c) 1.

Step 1: Identify the Base and Exponent

For part (a), the expression is \( \log_{5} 5^{4} \). We have a logarithm with base \(5\) and its argument is \(5^4\).

Step 2: Apply the Power Rule for Logarithms

According to the power rule, \( \log_{b} (b^x) = x \). Therefore, for \( \log_{5} 5^4 \), the answer is \(4\).

Step 3: Convert to a Common Base

In part (b), the expression is \( \log_{4} 64 \). First, express 64 as a power of 4: \(64 = 4^3\).

Step 4: Use the Power Rule Again

Now that we have \(64 = 4^3\), substitute this back into the logarithm to get \( \log_4 (4^3) = 3 \).

Step 5: Apply Identity Logarithm Property

For part (c), the expression is \( \log_{9} 9 \). Using the identity logarithm property that \( \log_b b = 1 \), since the base and the argument are the same, the result is \(1\).

Step 6: Compile Final Answers

The solutions to the individual expressions are: (a) 4, (b) 3, (c) 1.

Key Concepts

Logarithm Properties

Logarithms have specific properties that simplify calculations and help us understand relationships between numbers.
One fundamental property is that logarithms and exponents are inverses.
This means a logarithm can help find the exponent that a base must be raised to, to result in a specific number.

Here are some key properties:

• The Product Rule: States that the logarithm of a product equals the sum of the logarithms: \( \log_b (MN) = \log_b M + \log_b N \).
• The Quotient Rule: States that the logarithm of a quotient is the difference of the logarithms: \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \).
• The Power Rule: States that the logarithm of a power is the exponent times the logarithm: \( \log_b (M^n) = n \cdot \log_b M \).

These properties make it easier to break down complex expressions into simpler parts.
In our exercise, understanding these properties allowed us to solve for the exponents quickly and accurately.

Power Rule for Logarithms

The power rule for logarithms is incredibly helpful when dealing with expressions that include exponents.
This rule states that when you take a logarithm of a number that is raised to an exponent, you can multiply the exponent with the logarithm of the base number.

Mathematically, it can be written as:

• \( \log_b (M^n) = n \cdot \log_b M \)

For example, in part (a) of the exercise, we used the power rule to simplify \( \log_5 5^4 \).
According to the rule, the calculation becomes straightforward: \( 4 \cdot \log_5 5 \).
Since \( \log_5 5 \) equals 1 (as per the identity property), you simply multiply 4 by 1, simplifying it to 4.

Using this power rule is especially useful for simplifying expressions where the argument of the logarithm is a power of the base.
This saves time and effort in longer calculations.

Identity Logarithm Property

The identity logarithm property is a fundamental concept in logarithms.
This property states that if the base of the logarithm and the argument (the number you're taking the log of) are identical, then the logarithm equals 1.

Formally, this is written as:

• \( \log_b b = 1 \)

This simplifies understanding and calculations significantly.
For example, in part (c) of our exercise, we calculated \( \log_9 9 \).
Since the base and argument are the same, using the identity property directly gives us a value of 1.

Knowing this property allows us to quickly and efficiently determine results without needing further calculations.
It reaffirms the inversely proportional nature between logarithms and exponents, where an exponent that returns the base as a result is clearly 1.