SOLUTION: If log a^2 b^3 = x and log (a/b) = y, what are the values of log a and log b?

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: If log a^2 b^3 = x and log (a/b) = y, what are the values of log a and log b?      Log On


   

Question 955429: If log a^2 b^3 = x and log (a/b) = y, what are the values of log a and log b?
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
(1) +x+=+log%28+a%5E2%2Ab%5E3+%29+
(2) +y+=+log%28+a%2Fb+%29+
-------------------------
(1) +x+=+log%28+a%5E2+%29+%2B+log%28+b%5E3+%29+
(1) +x+=+2%2Alog%28a%29+%2B+3%2Alog%28b%29+
-------------------------------
(2) +y+=+log%28a%29+-+log%28b%29+
-------------------------
Multiply both sides of (2) by +3+
and add the equations
(1) +x+=+2%2Alog%28a%29+%2B+3%2Alog%28b%29+
(2) +3y+=+3%2Alog%28a%29+-+3%2Alog%28b%29+
----------------------------
+x+%2B+3y+=+5%2Alog%28a%29+
+log%28a%29+=+%28+x+%2B+3y+%29+%2F+5+
and
(2) +y+=+log%28a%29+-+log%28b%29+
(2) +y+=+%28+x+%2B+3y+%29+%2F+5+-+log%28b%29+
(2) +log%28b%29+=+%28+x+%2B+3y+%29+%2F+5+-+y+
(2) +log%28b%29+=+x%2F5+%2B+%283y%29%2F5+-+%285%2Ay%29%2F5+
(2) +log%28b%29+=+x%2F5+-+%282y%29%2F5+
----------------------------
+log%28a%29+=+%28+x+%2B+3y+%29+%2F+5+
+log%28b%29+=+%28+x+-+2y+%29+%2F+5+ answers
------------------------
check:
Let +a+=+10+
Let +b+=+100+
-----------------
(1) +x+=+log%28+a%5E2%2Ab%5E3+%29+
(1) +x+=+log%28+10%5E2%2A100%5E3+%29+
(1) +x+=+log%28+10%5E8+%29+
(1) +x+=+8+
and
(2) +y+=+log%28+a%2Fb+%29+
(2) +y+=+log%28+10%2F100+%29+
(2) +y+=+log%28+10%5E%28-1%29+%29+
(2) +y+=+-1+
----------------
+log%28a%29+=+%28+x+%2B+3y+%29+%2F+5+
+log%2810%29+=+%28+8+-+3+%29+%2F+5+
+log%2810%29+=+5%2F5+
+1+=+1+
OK
+log%28b%29+=+%28+x+-+2y+%29+%2F+5+
+log%28100%29+=+%28+8+-+2%2A%28-1%29+%29+%2F+5+
+2+=+%28+8%2B2+%29+%2F+5+
+2+=+10%2F5+
+2+=+2+
OK