Which log is "heavier"?

Algebra Level 1

log ⁡ 3 4 , log ⁡ 4 5 \large\log_3 4 \qquad , \qquad \log_4 5

Which of these numbers is larger in value?

Image Credit: Wikimedia IllestFlip .
log ⁡ 3 4 \log_{3} 4 log ⁡ 4 5 \log_{4} 5

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Hobart Pao by Hobart Pao , 23, USA

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10 solutions

Raj Rajput
Jul 14, 2015

10 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Is there a way to solve this without logarithmic tables?

Nice handwriting!

Mike Roscope - 5 years, 11 months ago
Blackpen Redpen
Jul 14, 2015

6 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Yes. This is the generalization via calculus.

For simplicity sake, can you solve this simply based on the fact that both functions f ( x ) = 3 x f(x) = 3^x and g ( x ) = 4 x g(x) = 4^x are concave upwards?

J Dotson
Sep 30, 2015

Awesome solutions.

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Rohan Naidu
Aug 5, 2015

Observe that the derivative of the function log x is proportional to 1/x, a monotonically decreasing function on positive reals. This means the rate at which log x is increasing is decreasing, that is, log x grows more between 3 and 4, than between 4 and 5. Therefore, log 4 is greater than log 3 by more than log 5 is greater than log 4...which leads to the desired result.

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汶良 林
Jul 29, 2015

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Chenjia Lin
Jul 25, 2015

My idea was to convert l o g 3 4 log_3 4 into 3 n = 4 3^{n} = 4 , and l o g 4 5 log_4 5 into 4 n = 5 4^{n} = 5 . Instead of finding n n in both equations, I plugged in 2 2 for the value of n. ignoring the right side of the equations. The result from plugging in 2 2 gave me a larger value for 4 n = 5 4^{n} = 5 .

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Lew Sterling Jr
Jul 16, 2015

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Tanishq Bhatia
Jul 16, 2015

Well, the way I saw this was that it is easier to increase a number by 1/4th than by 1/3rd.This means that the power to 4 to get 5 would be less than the power required for 3 to make it 4.

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Moderator note:

You can't do that with logarithms.

That was my reasoning also. Isn't it valid ?

Louis LeLouis - 5 years, 7 months ago
Ayan Bhuyan
Jul 15, 2015

Let 3^x=4 and 4^y = 5. So, x/y=log5/log3 => x>y.

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Moderator note:

Wrong. Check your working again.

Thiago Alves
Jul 15, 2015

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Check your working again. It's wrong.

Your approach is no different from interpreting the values from a logarithmic table.

Note: It is not "determinator", but "denominator" and it is positive.

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