Log questions

Algebra Level pending

If log ⁡ ( a − b ) = log ⁡ ( a 2 − b 2 ) \log(a-b)=\log(a^2-b^2) and it is given that b = 3 b=3 Find a a .

To clarify, we are only dealing with real numbers here.

no solution 0 1 -2

Animesh Pal by Animesh Pal , 30, India

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2 solutions

Using the difference of squares formula, we find that

l o g ( a − b ) log(a - b) = l o g ( a − b ) log(a - b) + l o g ( a + b ) log(a + b) .

Thus if there is a solution then l o g ( a + b ) = 0 log(a + b) = 0 .

This would then require that a + b = 1 a + b = 1 , i.e., that a = − 2 a = -2 .

But this would make a − b = − 5 a - b = - 5 , making l o g ( a − b ) log(a - b) meaningless. Thus there is no real solution.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

( a − b ) (a-b) must be a positive number. If b = 3 b=3 , then a a should be > 3. Thus, the answer is no solution.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Very good reasoning. Congratulations.

Niranjan Khanderia - 6 years, 11 months ago

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