Question
Solve the equation
log[14](x+49) - log[196] x=1
log[14](x+49) - log[196] x=1
Answers
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
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.