How to Solve a Logarithmic Equation with 3 Logs: log_a(x) + log_a(x - 2) = log_a(x + 4) | Summary and Q&A

TL;DR
The video explains how to solve logarithmic equations by combining logs using the product rule and factoring quadratic equations.
Key Insights
- 🧑💻 Combining logarithms using the product rule simplifies equations with multiple logs.
- ⚾ Exponentiating cancels out the logarithms by matching the base and the exponent.
- ❓ Quadratic equations obtained from logarithmic equations can be solved by factoring.
Transcript
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Questions & Answers
Q: How do you combine logarithms using the product rule?
The product rule states that the sum of two logarithms with the same base is equal to the logarithm of their product. In the equation, log base a of x + log base a of x - 2 can be combined to log base a of x(x - 2).
Q: Why is it important to exponentiate after combining the logarithms?
Exponentiating is done to cancel out the logarithms. By raising both sides of the equation to the power of the logarithm's base, the base and the logarithm will cancel out, leaving a simplified equation.
Q: How do you solve a quadratic equation obtained from a logarithmic equation?
To solve the quadratic equation, set it equal to zero and factor it. Find two numbers that multiply to give the constant term and add up to the coefficient of the linear term. These numbers will be the possible solutions for the equation.
Q: Why is it necessary to check the solutions when there are multiple logarithms?
The domain of the logarithm function consists of positive numbers only. Therefore, it is important to check if the solutions obtained satisfy this condition. In this case, the solution x = -1 is not valid, as it would result in taking the log of a negative number.
Summary & Key Takeaways
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The video demonstrates how to combine logarithms using the product rule to simplify an equation with multiple logs.
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Exponentiation is used to eliminate the logarithms by cancelling out the base.
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The quadratic equation obtained from the simplification is then factored to find the possible solutions.
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