Solve the equation

Question

log[14](x+49) - log[196] x=1

Answer 1

Given
log₁₄x+49 - log₁₉₆x = 1

We will use a lemma that
log_ax=log_a²=x²

Since 196=14², we write above as
log₁₄x+49 - log_14²x = 1
Then
log_14²(x+49)² - log_14²x = 1
Rewrite using laws of logarithm:
log_14²(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

Answer 2

We will use a lemma that states:
log_ax = log_a²x²

The proof of the lemma is left to you as an exercise, if it was not already covered in your course.