Asked by Alley
Solve the equation
log[14](x+49) - log[196] x=1
log[14](x+49) - log[196] x=1
Answers
Answered by
MathMate
Given
log<sub>14</sub>x+49 - log<sub>196</sub>x = 1
We will use a lemma that
log<sub>a</sub>x=log<sub>a²</sub>=x²
Since 196=14², we write above as
log<sub>14</sub>x+49 - log<sub>14²</sub>x = 1
Then
log<sub>14²</sub>(x+49)² - log<sub>14²</sub>x = 1
Rewrite using laws of logarithm:
log<sub>14²</sub>(x+49)²/x = 1
Using the alternate form from the definition of logarithms,
(x+49)^2/x = 14²
Transpose and solve for x to get
(x-49)²=0, or
x=49
log<sub>14</sub>x+49 - log<sub>196</sub>x = 1
We will use a lemma that
log<sub>a</sub>x=log<sub>a²</sub>=x²
Since 196=14², we write above as
log<sub>14</sub>x+49 - log<sub>14²</sub>x = 1
Then
log<sub>14²</sub>(x+49)² - log<sub>14²</sub>x = 1
Rewrite using laws of logarithm:
log<sub>14²</sub>(x+49)²/x = 1
Using the alternate form from the definition of logarithms,
(x+49)^2/x = 14²
Transpose and solve for x to get
(x-49)²=0, or
x=49
Answered by
MathMate
We will use a lemma that states:
log<sub>a</sub>x = log<sub>a²</sub>x²
The proof of the lemma is left to you as an exercise, if it was not already covered in your course.
log<sub>a</sub>x = log<sub>a²</sub>x²
The proof of the lemma is left to you as an exercise, if it was not already covered in your course.
There are no AI answers yet. The ability to request AI answers is coming soon!
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.