SOLUTION: Solve without using calculator. Which one is greater log3 to the base 2 or log5 to the base 3?

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Question 1107601: Solve without using calculator.
Which one is greater log3 to the base 2 or log5 to the base 3?

Found 3 solutions by Alan3354, math_helper, herosz:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Which one is greater log3 to the base 2 or log5 to the base 3?
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log%282%2C3%29 OR log%283%2C5%29
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convert both to the same base:
log%282%2C3%29+=+log%283%29%2Flog%282%29
log%283%2C5%29+=+log%285%29%2Flog%283%29
From there you can use a calculator.
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A better solution:
log%282%2C3%29+=+log%286%2C3%29%2Flog%286%2C2%29+=+log%286%2C1.5%29
log%283%2C5%29+=+log%286%2C5%29%2Flog%286%2C3%29+=+log%286%2C1.67%29
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Ooops. That's not right.
Another tutor did it. You OK with that solution?

Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
In the 'better solution' given by another tutor, the answer comes out incorrect.
log(a) / log(b) does not equal log(a/b) because log(a/b) = log(a)-log(b).
I still don't have a non-calculator answer, maybe tomorrow, its late.

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Non-calculator solution:
Comparing +log%282%2C%283%29%29+ vs +log%283%2C%285%29%29+ is the same as comparing x and y in:
+2%5Ex+=+3+ vs +3%5Ey+=+5+
Claim: x>3/2
Set x=3/2 to check:

Comparing the square of this to +3%5E2+ we get +8+%3C+9+ implying x > 3/2 (using 3/2 as the exponent gives a number that is slightly too small)

What if y = 3/2?

Comparing the square of this to +5%5E2++ we get +27+%3E+25+ implying y < 3/2 (using 3/2 as the exponent gives a number that is slightly too big)

We now have +y+%3C+%283%2F2%29+%3C+x so we can say +y+%3C+x+.
Therefore +log%283%2C%285%29%29+%3C+log%282%2C%283%29%29+
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Answer by herosz(4) About Me  (Show Source):