Problem 34 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 33-34: Solve the equation. $$ \log _{4} x+\log _{8} x=1 $$

Step-by-Step Solution

Answer

The solution is \( x = 2^{6/5} \).

Step 1: Apply Logarithm Product Property

To solve the equation \( \log _{4} x + \log _{8} x = 1 \), we'll first apply the property of logarithms: \( \log_{a} b + \log_{a} c = \log_{a}(b \cdot c) \). However, since the bases are different, we will use the change of base formula.

Step 2: Convert Logs to Common Base

Use the change of base formula: \( \log_{a} b = \frac{\log_{c} b}{\log_{c} a} \). Thus, \( \log_{4} x = \frac{\log_{2} x}{\log_{2} 4} = \frac{\log_{2} x}{2} \) and \( \log_{8} x = \frac{\log_{2} x}{\log_{2} 8} = \frac{\log_{2} x}{3} \). Substitute back into the original equation.

Step 3: Simplify Equation

Substitute and simplify the equation: \( \frac{\log_{2} x}{2} + \frac{\log_{2} x}{3} = 1 \). Combine the fractions: \( \frac{3 \cdot \log_{2} x + 2 \cdot \log_{2} x}{6} = 1 \), which simplifies to \( \frac{5 \log_{2} x}{6} = 1 \).

Step 4: Solve for \( \log_{2} x \)

To isolate \( \log_{2} x \), multiply both sides by 6: \( 5 \log_{2} x = 6 \). Then divide both sides by 5 to get \( \log_{2} x = \frac{6}{5} \).

Step 5: Solve for \( x \)

Convert from logarithmic form to exponential form: \( x = 2^{\frac{6}{5}} \). Simplify this expression to find the value of \( x \).

Key Concepts

Logarithm Product Property

The logarithm product property is a useful tool when dealing with logarithmic equations. It states that when you add two logarithms with the same base, you can combine them into a single logarithm by multiplying their arguments. The property can be written as \( \log_{a} b + \log_{a} c = \log_{a}(b \cdot c) \). This property plays a crucial role when trying to simplify logarithmic equations for easier solution.

In the given exercise, we initially consider applying this property. However, we notice that the logarithms have different bases (4 and 8), which makes direct application of the product property inapplicable right from the start. Instead, we need to equalize the bases to use similar properties, leading us to the change of base formula next.

Change of Base Formula

The change of base formula allows us to convert logarithms from one base to another, which is especially helpful when the base is not compatible with other terms in the equation. The formula is expressed as \( \log_{a} b = \frac{\log_{c} b}{\log_{c} a} \). It enables us to work with logarithms more flexibly by choosing a common base for easier manipulation and comparison.

In the exercise, we use this formula to convert \( \log_{4} x \) and \( \log_{8} x \) to base 2. This is because the logarithm with a base of 2 is easily manageable. Thus:

• \( \log_{4} x = \frac{\log_{2} x}{\log_{2} 4} = \frac{\log_{2} x}{2} \)
• \( \log_{8} x = \frac{\log_{2} x}{\log_{2} 8} = \frac{\log_{2} x}{3} \)

Substituting these into the equation allows us to continue the simplification process.

Exponential Form

The exponential form is another perspective of representing logarithmic expressions, facilitating the process of solving for unknown values. It is derived from the definition of a logarithm: if \( \log_{b} a = c \), then in exponential form it is \( b^{c} = a \). This transformation can help intuitively understand how to isolate variables or solve for them in equations.

After simplifying our logarithmic equation to \( \log_{2} x = \frac{6}{5} \), we can use the exponential form to find \( x \). By applying the definition, we switch to exponential form by interpreting \( \log_{2} x \) as stating that \( x = 2^{\frac{6}{5}} \). Changing to exponential form thus paves the way to solve explicitly for our variable.

Logarithmic Equations

Logarithmic equations are equations that involve logarithms and often require specific techniques to solve, such as the use of logarithmic properties, conversion to exponential form, and simplification steps. The goal is to manipulate the equation until the variable of interest is isolated and solved.

In the problem at hand, we encounter a logarithmic equation \( \log_{4} x + \log_{8} x = 1 \). To solve such equations, we apply strategies like:

• Using logarithm properties (though initially inapplicable here without equal bases).
• Employing the change of base formula for consistency.
• Transforming logarithmic expressions into exponential form for final resolution.

Holistically, solving logarithmic equations involves understanding and applying these concepts carefully to arrive at a solution efficiently. The final answer in this case, \( x = 2^{\frac{6}{5}} \), showcases the culmination of these methods.