Problem 34 - Algebra A Combined Function

Question

Find the value of each logarithmic expression. $$ \log _{8} \frac{1}{2} $$

Step-by-Step Solution

Answer

The value of \( \log_8 \left( \frac{1}{2} \right) \) is \( \frac{-1}{3} \).

Step 1: Set up the equation

The problem is to find the value of \( \log_8 \left( \frac{1}{2} \right) \). To start, let's set \( x = \log_8 \left( \frac{1}{2} \right) \). This means we need to solve for \( x \) in the following equation: \( 8^x = \frac{1}{2} \).

Step 2: Express as exponents

We know that \( 8 = 2^3 \), so substituting we have \( (2^3)^x = \frac{1}{2} \). This can be rewritten as \( 2^{3x} = \frac{1}{2} \).

Step 3: Set exponents equal

The right side can be rewritten as a power of 2: \(\frac{1}{2} = 2^{-1}\). So, the equation \( 2^{3x} = 2^{-1} \) implies that the exponents must be equal: \( 3x = -1 \).

Step 4: Solve for x

To solve for \( x \), divide both sides by 3: \( x = \frac{-1}{3} \).

Key Concepts

Change of Base Formula

The change of base formula is a tool used in logarithms that helps to convert a logarithm with any base to a logarithm with base 10 or base e (natural logarithm). This can be particularly helpful if you're dealing with uncommon bases or if your calculator only offers functions for base 10 or base e logs.
Logarithmic expressions, such as \( \log_b a \), can be rewritten with a different base using the formula:
\[\log_b a = \frac{\log_c a}{\log_c b}\]
where \( c \) is the new base you want to use (often 10 or \( e \) since calculators have these functions available). Utilizing the change of base formula is a critical skill in simplifying complex logarithmic expressions and making calculations more feasible when the original base is not easily manageable.

Exponential Equations

Exponential equations are equations where variables appear as exponents. The key to solving these equations often involves expressing every term in the equation as a power of the same base. In the context of the original problem, this meant expressing both sides of the equation using the base 2.
In our example:
* Start with \( 8^x = \frac{1}{2} \).* Recognize that 8 can be rewritten as \( 2^3 \).* Rewrite the equation as \( (2^3)^x = 2^{-1} \), leading to \( 2^{3x} = 2^{-1} \).
By setting the base the same on both sides, we significantly simplify the problem to just comparing the powers, leading to an equation solely in terms of exponents, which is easier to solve. This approach is powerful for unraveling exponential relationships.

Solving Logarithmic Expressions

Solving logarithmic expressions involves finding the unknown value that satisfies the given logarithmic equation. The process typically starts with rewriting the logarithmic equation in its equivalent exponential form, which can simplify the path to a solution.
In the initial exercise, we set \( x = \log_8 \left( \frac{1}{2} \right) \) which led us to work with \( 8^x = \frac{1}{2} \). By expressing both sides as powers of 2, we then had \( 2^{3x} = 2^{-1} \). Since the bases were equal, we could focus on the exponents \( 3x = -1 \).
* Solving for \( x \), we divided both sides by 3, resulting in \( x = \frac{-1}{3} \).
This process highlights one of the core methods for handling logarithmic expressions: translating them into exponential relationships and then applying straightforward algebraic techniques to solve.