- #1

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## Homework Statement

## Homework Equations

## The Attempt at a Solution

I assume I can't use a calculator obviously.. so I'm quite stuck. The answer is 5, but I have no idea how to get that.

(log

_{2}3)(log

_{3}4)(log

_{4}5) ... (log

_{31}32)

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- Thread starter feihong47
- Start date

- #1

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I assume I can't use a calculator obviously.. so I'm quite stuck. The answer is 5, but I have no idea how to get that.

(log

- #2

Mute

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$$\log_b a = \frac{\log_c a}{\log_c b}$$

for any positive, real numbers a, b and c (with c > 1).

- #3

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- #4

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You use LaTeX, that is, the [ itex ][ /itex ] -tags (remove the spaces) . There's a guide in the forums somewhere if you're not familiar with it, I'll edit a link to it to this post if I find it. You can also open the "LaTeX Reference" by clicking the Σ in the toolbar.

If you don't have to write anything complicated, you can just use the quick symbols and x

EDIT: Here's the LaTeX guide.

- #5

SammyS

Staff Emeritus

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You could also approach this as follows:## Homework Statement

## Homework Equations

## The Attempt at a Solution

I assume I can't use a calculator obviously.. so I'm quite stuck. The answer is 5, but I have no idea how to get that.

(log_{2}3)(log_{3}4)(log_{4}5) ... (log_{31}32)

Let [itex]\displaystyle y=(\log_{2}3)\,(\log_{3}4)\,(\log_{4}5)\,\dots\,( \log_{31}32) [/itex]

Then, [itex]\displaystyle 2^y=2^{(\,(\log_{2}3)\,(\log_{3}4)\,(\log_{4}5)\, \dots\,( \log_{31}32)\,)} [/itex]

By laws of exponents and the definition of a logarithm,

[itex]2^{(\,(\log_{2}3)\,(\log_{3}4)\,(\log_{4}5)\, \dots\,( \log_{31}32)\,)}[/itex]

[itex]=\left(2^{(\log_{2}3)}\right)^{(\,(\log_{3}4)\,( \log_{4}5)\, \dots\,( \log_{31}32)\,)}[/itex]

[itex]=3^{(\,(\log_{3}4)\,( \log_{4}5)\, \dots\,( \log_{31}32)\,)}[/itex]

[itex]=\left(3^{(\log_{3}4)}\right)^{(\,( \log_{4}5)\, \dots\,( \log_{31}32)\,)}[/itex]

[itex]=4^{(\,( \log_{4}5)\, \dots\,( \log_{31}32)\,)}[/itex]

etc.

[itex]=\left(31^{(\log_{31}32)}\right)[/itex]

[itex]=32[/itex]

[itex]=3^{(\,(\log_{3}4)\,( \log_{4}5)\, \dots\,( \log_{31}32)\,)}[/itex]

[itex]=\left(3^{(\log_{3}4)}\right)^{(\,( \log_{4}5)\, \dots\,( \log_{31}32)\,)}[/itex]

[itex]=4^{(\,( \log_{4}5)\, \dots\,( \log_{31}32)\,)}[/itex]

etc.

[itex]=\left(31^{(\log_{31}32)}\right)[/itex]

[itex]=32[/itex]

- #6

HallsofIvy

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- #7

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Very nice approach. Thanks!

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