SOLUTION: Show that logb (a) loga (b) = 1

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Question 1089580: Show that logb (a) loga (b) = 1
Found 3 solutions by Fombitz, math_helper, MathTherapy:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
log%28a%2C%28b%29%29=log%28%28b%29%29%2Flog%28%28a%29%29
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log%28b%2C%28a%29%29=log%28%28a%29%29%2Flog%28%28b%29%29
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Take it from there.

Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
Say +X+=+log%28a%2C%28b%29%29+

Raise a to both sides:
+a%5EX+=+a%5E%28log%28a%2C%28b%29%29%29+
The right hand side is just b:
+a%5EX+=+b+
Now take log base b of both sides:
+log%28b%2C%28a%5EX%29%29+=+log%28b%2C%28b%29%29+
The left hand side is +X%2Alog%28b%2C%28a%29%29+ while the right hand side is 1:
++X%2Alog%28b%2C%28a%29%29+=+1+ —> +log%28b%2C%28a%29%29%2AX+=+1+
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Finally, since +X+=+log%28a%2C%28b%29%29+, the proof is completed by substituting for X:
+highlight%28+log%28b%2C%28a%29%29%2Alog%28a%2C%28b%29%29+=+1%29+

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

Show that logb (a) loga (b) = 1
PROVE: matrix%281%2C3%2C+log+%28b%2C+%28a%29%29+%2A+log+%28a%2C+%28b%29%29%2C+%22=%22%2C+1%29
%28log+%28%28a%29%29%2Flog+%28%28b%29%29%29+%2A+%28log+%28%28b%29%29%2Flog+%28%28a%29%29%29 ------- Applying CHANGE of BASE

That's how SIMPLE this is.