Problem 16 - Algebra and Trigonometry

Question

\(7-18\) Evaluate the expression. $$ \log _{2} 8^{33} $$

Step-by-Step Solution

Answer

The value of the expression is 99.

Step 1: Recognize the Base Change in Expression

The given expression is \( \log_{2} 8^{33} \). Notice that \( 8 \) is a power of \( 2 \). Specifically, \( 8 = 2^3 \). This allows us to rewrite the expression with base 2.

Step 2: Rewrite the Expression Using Powers of Base

Replace \( 8 \) with \( 2^3 \) in the expression: \[\log_{2} (2^3)^{33}\]Using the property of exponents, \( (a^m)^n = a^{m \cdot n} \), rewrite it as:\[\log_{2}(2^{3 \times 33})\]

Step 3: Simplify the Expression Further

Calculate the exponent: \( 3 \times 33 = 99 \). The expression now becomes:\[\log_{2} (2^{99})\]This matches the form \( \log_{a} (a^b) \).

Step 4: Apply Logarithmic Identity

Use the identity \( \log_{a} (a^b) = b \). For our expression, this means:\[\log_{2} (2^{99}) = 99\]

Step 5: State the Final Result

Thus, the simplified evaluation of the logarithmic expression is 99.

Key Concepts

Exponent Rules

Understanding how exponents work is crucial for simplifying expressions like \(\log_{2} 8^{33}\). Exponent rules help us manipulate and transform expressions for easier evaluation. Here are some key rules:

• Power of a Power rule: When you have a power raised to another power, \((a^m)^n\), you multiply the exponents. So, \((2^3)^{33} = 2^{3 \times 33} = 2^{99}\).
• Product of Powers rule: When multiplying two powers with the same base, \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers rule: When dividing powers with the same base, \(a^m / a^n = a^{m-n}\).

Exponent rules are foundational for dealing with powers in expressions. They help transform complex forms into simpler, comparable ones. For our original expression, using exponent rules enables factorization and simplification which pave the way for easier computation using log properties.

Logarithmic Identities

Logarithmic identities are powerful tools for simplifying expressions and solving equations involving logarithms. For our problem, one such identity is particularly helpful:

• Logarithm of a Power: \(\log_{a} (a^b) = b\). This identity tells us when the base of the logarithm (2 in this scenario) matches the base of the exponent (also 2), the logarithm simplifies to the exponent itself. Thus, in \(\log_{2} (2^{99})\), the result is simply 99.
• Product Rule: \(\log_{a} (xy) = \log_{a} x + \log_{a} y\).
• Quotient Rule: \(\log_{a} (x/y) = \log_{a} x - \log_{a} y\).

Utilizing these identities makes the complex expressions handleable. By applying the Logarithm of a Power identity, what starts as a seemingly intricate calculation simplifies to a straightforward retrieval of the exponent value.

Base Change in Logarithms

Changing the base of numbers within an expression can make the expression much simpler to work with, particularly when dealing with logarithms. In the original problem, base change plays a critical role. The number 8 can be rephrased as a power of 2:

• Base Conversion: Since 8 is equivalent to \(2^3\), we replace 8 with \(2^3\) in the original expression \(\log_{2} 8^{33}\).
• By changing the expression to \(\log_{2} (2^3)^{33}\), the expression becomes easier to deal with, as the base of the logarithm matches the base of the expression.

This base conversion is a powerful move that simplifies calculations. When the expression \(\log_{2} (2^{99})\) appears, it directly makes use of a logarithmic identity for a quick solution. Understanding how and when to change bases is a key skill in simplifying logarithmic functions efficiently.