8A F11 Q16

Question: Solve. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log_6(x+2)+\log_6(x-3) = 1 }

Foundations
1) How do we combine the two logs?
2) How do we remove the logs?
Answer:
1) One of the rules of logarithms says that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log(x) + \log(y) = \log(xy)}
2) The definition of logarithm tells us that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log_6(x) = y } , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6^y = x }

Solution:

Step 1:
Using a rule of logarithms the left hand side is equal to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log_6( (x + 2)(x - 3)}

Step 2:

By the definition of logarithms Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log_6( (x + 2)(x - 3) = 1} means Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6 = (x + 2)(x - 3)}

Step 3:
Now we do some arithmetic to solve for x. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 = (x + 2)(x - 3) - 6 = x^2 - x - 12 = (x - 4)(x + 3) } . So there are two possible answers.

Step 4:
We have to make sure the answers make sense in the context of the problem. Since the domain of the log function is $(0,\infty )$,    -3 is removed as a potential answer.

Final Answer:
x = 4.

Return to Sample Exam