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Magazine Chapter 6: Problem 80
Discuss how to solve the given equation. Carry out your ideas. $$ \log _{3} x-\log _{6} x-2=0 $$
Short Answer
Expert verified
Solve \( x = 3^{\frac{2 \times \log_3 6}{\log_3 6 - 1}} \).
Step by step solution
01
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.
02
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: \( \log_3 x - \frac{\log_3 x}{\log_3 6} - 2 = 0 \).
03
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: \( \log_3 x (\frac{\log_3 6 - 1}{\log_3 6}) = 2 \).
04
Solve for \( \log_3 x \)
Isolate \( \log_3 x \) by multiplying both sides by \( \log_3 6 \) and dividing by \( \log_3 6 - 1 \): \( \log_3 x = \frac{2 \times \log_3 6}{\log_3 6 - 1} \).
05
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
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 \( x \).
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 \( x \).
