# simplify natural log times natural log

• Mar 3rd 2010, 07:11 PM
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simplify natural log times natural log
So i have this problem that i am supposed to simplify to a single logarithm or single exponential.

Log3(5) * Log5(3)

So far i have
ln(5/3) * ln(3/5)

how do i simplify this even more? is there a rule for multiplying two natural logs? thanks!
• Mar 3rd 2010, 07:29 PM
pickslides
Are you sure this is correct?

I get

$\log_3(5) = \frac{\ln(5)}{\ln(3)}$

and

$\log_5(3) = \frac{\ln(3)}{\ln(5)}$

therefore

$\log_3(5) \times \log_5(3) = \frac{\ln(5)}{\ln(3)}\times \frac{\ln(3)}{\ln(5)} = 1$
• Mar 3rd 2010, 09:31 PM
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Quote:

Originally Posted by pickslides
Are you sure this is correct?

I get

$\log_3(5) = \frac{\ln(5)}{\ln(3)}$

and

$\log_5(3) = \frac{\ln(3)}{\ln(5)}$

therefore

$\log_3(5) \times \log_5(3) = \frac{\ln(5)}{\ln(3)}\times \frac{\ln(3)}{\ln(5)} = 1$

ohh okay that makes sense. Thanks!