SOLUTION: Please help me. How to prove that log b x- log b y = log b (x/y). (* "b" should be smaller letter) Thank you.

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: Please help me. How to prove that log b x- log b y = log b (x/y). (* "b" should be smaller letter) Thank you.       Log On


   

Question 1113137: Please help me. How to prove that log b x- log b y = log b (x/y). (* "b" should be smaller letter)
Thank you.
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
The property of exponents +x%5Ea%2Fx%5Eb++=+x%5E%28a-b%29+ (*) will be used. This can be easily proved.
Let log%28b%2C%28x%2Fy%29%29+=+w+ <—> +b%5Ew+=+x%2Fy+ (1)
Let +log%28b%2C%28x%29%29+=+v+ <—> +b%5Ev+=+x+ (2)
Let +log%28b%2C%28y%29%29+=+z+ <—> +b%5Ez+=+y+ (3)
+b%5Ew+=+x%2Fy+=+b%5Ev%2Fb%5Ez+=+b%5E%28v-z%29+ (property (*) used on last step)
Picking off just this part:
+x%2Fy+=+b%5E%28v-z%29+
and taking log base b of both sides:
+log%28b%2C%28x%2Fy%29%29+=+log%28b%2C%28b%5E%28v-z%29%29%29+=+v-z+
Substituting (2) and (3) for v and z, respectively, completes the proof:
+log%28b%2C%28x%2Fy%29%29=+log%28b%2C%28x%29%29+-+log%28b%2C%28y%29%29+