Problem 24 - Precalculus with Calculus Previews

Question

Solve the given logarithmic equation. $$ 3 \log _{8} x=\log _{8} 36+\log _{8} 12-\log _{8} 2 $$

Step-by-Step Solution

Answer

The solution is \( x = 6 \).

Step 1: Apply Logarithmic Properties to the Right Side

We start by using the properties of logarithms to simplify the right side of the equation \(3 \log_{8} x = \log_{8} 36 + \log_{8} 12 - \log_{8} 2\). Use the product property which states \( \log_b M + \log_b N = \log_b (MN) \), and then the quotient property \( \log_b M - \log_b N = \log_b \left( \frac{M}{N} \right) \).Thus, \( \log_{8} 36 + \log_{8} 12 - \log_{8} 2 \) becomes:\[ \log_{8} \left( \frac{36 \times 12}{2} \right) \].

Step 2: Simplify the Expression on the Right Side

Calculate the expression inside the logarithm from Step 1:\[ \frac{36 \times 12}{2} = 216 \].Thus, the right side of the equation simplifies to \( \log_{8} 216 \). The equation now reads:\[ 3 \log_{8} x = \log_{8} 216 \].

Step 3: Remove the Logarithms by Taking the Exponent on Both Sides

From \( 3 \log_{8} x = \log_{8} 216 \), use the property that if \( a \log_b c = \log_b d \), then \( c^a = d \). Therefore:\[ x^3 = 216 \].

Step 4: Solve for x by Taking the Cube Root

To solve \( x^3 = 216 \), take the cube root of both sides:\[ x = \sqrt[3]{216} = 6 \].

Key Concepts

Logarithmic Properties

Logarithmic properties are essential tools in solving logarithmic equations. They help simplify complex logarithmic expressions, making them easier to solve. There are several key properties to remember:

• Product Property: This property states that the logarithm of a product is the sum of the logarithms. Mathematically, it is expressed as \( \log_b (MN) = \log_b M + \log_b N \).
• Quotient Property: For the division of two numbers, the logarithm can be expressed as the subtraction of their logarithms: \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \).

In our exercise, these properties are used to condense the terms on the right side of the equation from three distinct logarithms into one: \( \log_{8} 36 + \log_{8} 12 - \log_{8} 2 \) simplifies to \( \log_{8} \left( \frac{36 \times 12}{2} \right) \). This simplification sets the stage for further manipulations.

Exponential Equations

Exponentiation is the inverse operation of taking logarithms. Often, solving a logarithmic equation leads to dealing with exponential forms. Once the equation \( 3 \log_{8} x = \log_{8} 216 \) is simplified, we employ the property that if \( a \log_b c = \log_b d \), then \( c^a = d \). This step is crucial because it transforms the logarithmic format into an exponential one: \( x^3 = 216 \). Here, the variable \( x \) is raised to a power, which is a typical scenario in exponential equations. These equations often require operations like taking roots to solve for the base, as shown in the following steps.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying terms to make the equation easier to solve. It requires a clear understanding of mathematical operations and properties. In our equation, after applying logarithmic properties, the use of exponentiation allows us to create a straightforward expression: turning \( 3 \log_{8} x = \log_{8} 216 \) into \( x^3 = 216 \). To solve for \( x \), algebraic manipulation continues by taking the cube root of both sides. Thus, \( x = \sqrt[3]{216} \). This step-by-step breaking down of the problem illustrates the power of algebraic skills in simplifying and solving equations.

Educational Mathematics

Educational mathematics brings clarity to the various abstract concepts in solving equations, like logarithmic and exponential equations. It focuses on step-by-step processes, ensuring comprehensive understanding. The primary goal is to demystify mathematical operations through repeated practice with problems like these.

• Understanding Properties: Grasping the basic properties of logarithms and exponents lays the groundwork for tackling complex equations.
• Step-by-step Simplification: Breaking down a problem into smaller, manageable steps helps in maintaining focus and accuracy.
• Visualization and Connection: Connecting how properties transform an equation gives mathematical insights and builds confidence in problem-solving.

This exercise embodies educational mathematics by guiding students through a logical progression from recognizing logarithmic properties to solving the equation with algebraic techniques.