#1. What would be Extraneous Solution to log_6(x) + log_6(x + 1) = 1?

TL;DR

Learn how to solve logarithmic equations by combining logs and using the product rule, and identify extraneous solutions that do not check in the original equation.

Transcript

number one what would be the extraneous solution that you would obtain when you solve the given equation so we have log base 6 of X plus log base 6 of X plus 1 and that's all equal to 1 so the extraneous solution is the answer we get when we solve this problem that does not check so in order to get the answer we have to first solve the problem and ... Read More

Key Insights

• 🧑‍💻 Combining logs using the product rule is essential to simplify logarithmic equations.
• ⚾ Exponentiating with a base equal to the log's base eliminates the logarithm.
• 🧑‍🏭 Quadratic equations can be factored to find their solutions.

Questions & Answers

Q: How do you solve logarithmic equations when there are two logs with a plus sign?

To solve such equations, use the product rule to combine the logs into a single log. Then, exponentiate with a base equal to the log's base to eliminate the logarithm.

Q: What should be done if the base of the logarithm is different?

If the base is different, exponentiate with a base equal to the log's base to cancel out the log. This will give you a quadratic equation to solve.

Q: How do you simplify the quadratic equation obtained from the logarithmic equation?

Simplify the quadratic equation by distributing and combining like terms. Then, set the equation equal to zero to solve for the variable.

Q: How do you identify extraneous solutions in logarithmic equations?

After finding the solutions, plug each solution back into the original equation. If any solution results in an invalid input (e.g., negative numbers plugged into a logarithm), it is an extraneous solution.

Summary & Key Takeaways

• Logarithmic equations can be solved by combining logs using the product rule.

• To eliminate the logarithm, exponentiate with a base equal to the base of the logarithm.

• Quadratic equations can be obtained by simplifying the equation, and it can be factored to find the solutions.