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Questions and Answers
What is the first step in solving the equation $ rac{ log(x+6) }{ log x} = 2$?
What is the quadratic equation obtained after simplifying $log(x+6) = 2log x$?
Which of the following is a valid solution for the equation based on the logarithm rules?
What method is used to factor the quadratic equation $x^2 - x - 6 = 0$?
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Why is $x = -2$ not a permissible solution in the context of logarithms?
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Study Notes
Equation Overview
- The equation to solve is $\frac{ log(x+6) }{ log x} = 2$.
- The goal is to isolate $x$ using properties of logarithms.
Steps to Solve
- Start by rewriting the equation: $log(x+6) = 2log x$.
- Apply the logarithm power rule: $log(x+6) = log x^2$.
Application of Logarithmic Equality
- Use the equality rule of logarithms, which states if $log m = log n$, then $m = n$.
- Set $x+6 = x^2$.
Forming a Quadratic Equation
- Rearrange to form a quadratic equation: $x^2 - x - 6 = 0$.
- Factor the quadratic: $(x-3)(x+2) = 0$.
Finding Solutions
- The solutions from factoring are $x = 3$ and $x = -2$.
- Evaluate the validity of solutions by considering the logarithmic domain.
Logarithmic Domain Considerations
- $log x = log 3$ is valid, as $log$ of a positive number is defined.
- $log x = log(-2)$ is not valid, as logarithms of negative numbers are undefined in real numbers.
Final Solution
- The only permissible solution to the initial equation is $x = 3$.
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Description
This quiz covers the solution to the logarithmic equation involving logarithmic properties and equality rules. Participants will review critical steps in solving equations involving logs, specifically focusing on the provided example. Test your understanding of logarithms and their applications in algebra.
