Problem 80 - Precalculus with Calculus Previews

Question

Discuss how to solve the given equation. Carry out your ideas. $$ \log _{3} x-\log _{6} x-2=0 $$

Step-by-Step Solution

Answer

Solve \( x = 3^{\frac{2 \times \log_3 6}{\log_3 6 - 1}} \).

Step 1: Introduce the Change of Base Formula

To begin solving the equation, apply the change of base formula to express all logarithms to have a consistent base. The change of base formula is \( \log_a b = \frac{\log_c b}{\log_c a} \). We will convert \( \log_6 x \) to base 3 first.

Step 2: Rewrite the Equation

Using the change of base formula from Step 1, rewrite \( \log_6 x \) in terms of base 3: \( \log_6 x = \frac{\log_3 x}{\log_3 6} \). Substitute this into the original equation to get: \( \log_3 x - \frac{\log_3 x}{\log_3 6} - 2 = 0 \).

Step 3: Simplify the Equation

Simplify the equation by finding a common denominator for the logarithmic terms: \( \log_3 x (1 - \frac{1}{\log_3 6}) - 2 = 0 \). This results in: \( \log_3 x (\frac{\log_3 6 - 1}{\log_3 6}) = 2 \).

Step 4: Solve for \( \log_3 x \)

Isolate \( \log_3 x \) by multiplying both sides by \( \log_3 6 \) and dividing by \( \log_3 6 - 1 \): \( \log_3 x = \frac{2 \times \log_3 6}{\log_3 6 - 1} \).

Step 5: Calculate \( x \)

Since \( \log_3 x = y \) implies \( x = 3^y \), calculate \( x = 3^{\frac{2 \times \log_3 6}{\log_3 6 - 1}} \). This simplifies to a numerical computation to find the exact value.

Key Concepts

Change of Base Formula

When solving equations with logarithms, it's often useful to have a consistent base for all terms. This is where the change of base formula comes in handy. It allows you to convert a logarithm of any base into a logarithm with a desired base. The formula is given as \[ \log_a b = \frac{\log_c b}{\log_c a} \]Here, you can convert the logarithm of base \( a \) to a logarithm of base \( c \) by dividing the log of \( b \) in base \( c \) by the log of \( a \) in the same base.

• This formula is particularly useful when you're dealing with complex equations and need uniformity to simplify calculations.
• For our problem, we converted \( \log_6 x \) to a base 3 logarithm, ensuring all terms in the equation use the same base.

This conversion simplifies the equation, making it much easier to manipulate and solve. Converting between bases is a powerful tool when dealing with logarithmic equations.

Logarithm Properties

Logarithm properties are fundamental when working with logarithmic equations, enabling simplifications and manipulations. Here are some key properties to remember:

• Product Rule: \( \log_b (mn) = \log_b m + \log_b n \) - This helps combine or split logs.
• Quotient Rule: \( \log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n \) - Useful for handling divisions.
• Power Rule: \( \log_b (m^n) = n \cdot \log_b m \) - Assists in handling exponents.
• Change of Base: Already discussed, allows changing the base of logs.

In solving our problem, the change of base formula is a key property. However, understanding all properties is crucial as they often interplay in solving complex logarithmic equations.

Solving Equations

Solving logarithmic equations involves several steps, transforming the original problem into a simpler form. After converting all logarithms to the same base using the change of base formula, the equation becomes easier to manage.For our example:

• First, rewrite \( \log_6 x \) to \( \frac{\log_3 x}{\log_3 6} \) ensuring all logs are base 3.
• Combine terms and simplify using algebraic methods, finding a common denominator if needed.
• In the simplified form, isolate \( \log_3 x \). This often involves multiplying or dividing terms to set the equation in the form where the log equals a numerical value.
• Finally, solve for \( x \), turning the logarithmic equation back to exponential form to find the actual value of \( x \).

This process exemplifies the blend of algebraic manipulation and properties of logarithms to efficiently solve equations.

Numerical Computation

The final step in solving a logarithmic equation often involves numerical computation. Once the logarithmic form is converted to an exponential equation, you need to calculate the actual number.Given our equation transforms to: \[ x = 3^{\frac{2 \times \log_3 6}{\log_3 6 - 1}} \]To find \( x \), you

• Compute the numerical value of \( \log_3 6 \), typically using a calculator or logarithm table.
• Then calculate the entire exponent using the computed values.
• Finally, determine \( 3 \) raised to that computed exponent, providing the value for \( x \).

This computation highlights the practical application of logarithms to arrive at a specific numerical solution, rounding where necessary. Understanding how to perform numerical computations ensures accurate solutions when theoretical manipulations are complete.