Problem 9 - Algebra and Trigonometry

Question

Evaluate the expression. $$ \log _{4} 16^{100} $$

Step-by-Step Solution

Answer

The value is 200.

Step 1: Understand the Expression

The problem is to evaluate \( \log_{4} 16^{100} \). It first requires simplifying inside the logarithm before applying logarithmic rules.

Step 2: Simplify the Argument

Notice that 16 can be expressed as a power of 4: \( 16 = 4^2 \). So, \( 16^{100} = (4^2)^{100} \).

Step 3: Apply the Power of a Power Rule

Using the rule \( (a^m)^n = a^{m \cdot n} \), simplify \( (4^2)^{100} \) to \( 4^{200} \).

Step 4: Apply the Logarithm Rule

Use the logarithmic rule \( \log_b (b^a) = a \). Apply this here: \( \log_{4} 4^{200} = 200 \).

Key Concepts

Power of a Power Rule

The "Power of a Power Rule" is a concept used when dealing with exponents, specifically when you have an exponent raised to another exponent. It provides a straightforward way to simplify expressions efficiently and is vital in both algebra and calculus.
When you encounter an expression like \((a^m)^n\), the Power of a Power Rule allows you to combine these exponents by multiplying them together. This simplifies to \(a^{m \cdot n}\).
Let's take an example:

• Consider \((3^2)^4\). Using the Power of a Power Rule, you multiply 2 (the exponent of 3) by 4 (the outer exponent). This gives us \(3^{2 \cdot 4} = 3^8\).
• Applying this to the original exercise problem: \((4^2)^{100}\) simplifies to \(4^{200}\) using this rule. The inside exponent (2) is multiplied by the outer exponent (100), resulting in \(4^{200}\).

This rule helps to achieve much more manageable exponents and is a critical part of solving problems involving multiple layers of exponents.

Logarithmic Rules

Logarithmic rules are powerful tools in algebra that allow you to manipulate logarithmic expressions for simplification or evaluation. Among these rules, the ability to simplify using the base and exponent is crucial.
A key logarithmic rule states that \(\log_b (b^a) = a\). This rule makes finding the logarithm of a number that is a power of its own base incredibly straightforward.

• For instance, \(\log_3 (3^5) = 5\). Here, the base (3) and the power base (3) are the same, so the output is simply the exponent (5).
• In the provided exercise, the expression \(\log_4 4^{200}\) takes advantage of this rule. Since the base of the logarithm (4) and the base of the power (also 4) are identical, the logarithmic function "cancels" itself out, leaving only 200 as the answer.

Mastering these rules means you can solve many seemingly complicated logarithmic problems with just a few steps, making logarithms much less daunting.

Exponents

Exponents are a fundamental concept in mathematics, useful for expressing repeated multiplication concisely. Understanding how to work with exponents is necessary for advancing in various branches of mathematics.
The expression \(a^n\) refers to the number "a" being multiplied by itself "n" times. Here are some basic properties:

• Multiplying powers with the same base: \(a^m \cdot a^n = a^{m+n}\)
• Division of powers with the same base: \(a^m / a^n = a^{m-n}\)
• Power of a power: \((a^m)^n = a^{m \cdot n}\)

The exercise provided offers a practical example of using these principles. Initially, the number 16 is expressed as \(4^2\). When raised to the power of 100, it transforms into \((4^2)^{100}\), which simplifies to \(4^{200}\) using the Power of a Power Rule. This conversion is essential for solving the related logarithm problem.
Becoming fluent in exponent rules will greatly aid your problem-solving abilities, letting you tackle complex equations more confidently.