Logarithm problem

**Russellboy** · February 26th 2009, 10:54 PM

If log B base A = log A base B, then what is the value of (A)(B)? A and B are positive real numbers, neither A nor B is 1, and neither are equal.

**earboth** · February 27th 2009, 12:12 AM

Originally Posted by Russellboy

If log B base A = log A base B, then what is the value of (A)(B)? A and B are positive real numbers, neither A nor B is 1, and neither are equal.

Use the base change formula:

$\log_b(a)=\dfrac{\ln(a)}{\ln(b)}$

Therefore:

$\log_b(a)=\log_a(b)~\implies~ \dfrac{\ln(a)}{\ln(b)} = \dfrac{\ln(b)}{\ln(a)}~ \implies~(\ln(a))^2=(\ln(b))^2$

$(\ln(a))^2=(\ln(b))^2~\implies~(\ln(a))^2-(\ln(b))^2=0~\implies~$ $\left(\ln(a)-\ln(b)\right) \left(\ln(a)+\ln(b)\right)=0$

$\ln(a)=\ln(b)~\implies~ a= b$ (this case is excluded!) or

$\ln(a)=-\ln(b)~\implies~ a= \dfrac1b$

Therefore:

$\boxed{a \cdot b = 1}$

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