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Is there a general identity for multiplying or dividing logs with different bases and values?

$$\log_a(b)*\log_c(d)=\log_?(?)$$ $$\log_a(b)/\log_c(d)=\log_?(?)$$

I'm hoping for something that looks like the change of base identity below (except where bases don't match):

$$\log_x(a)/\log_x(b)=\log_b(a)$$

I realize you can use the change of base identity to get to an answer, but is there a way to do that without trying to change the bases to match?

EDIT For reference: I've found the following to work, but it seems like a bit of a hack and doesn't really help computation:

$$\log_a(b)*\log_c(d)=\log_\sqrt[\log_c(d)]{\,a}(b)$$ $$\log_a(b)/\log_c(d)=\log_a^{\log_c(d)}(b)$$

Is there a better way to do this?

Thanks!

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  • $\begingroup$ You can always cludge something $\log_m n =\log_{m^k} n^{\frac 1k}$ so $\log_a b\log_c d = \log_a b\log_b d^{\log_c b}= \log_a d^{\log_c b} = \log_c b*\log_a d$.... Viciously spinning your wheels but that's a neat result, that $\log_a b*\log_c d= \log_a d*\log_c b$. ... Wait, is that even true? $\endgroup$ – fleablood May 18 at 20:37
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No, there is no such identity, unless you count something like

$$ \log_a(b) \log_c(d) = \frac{\ln(b) \ln(d)}{\ln(a) \ln(c)} $$

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Using change of base, say for base $e$, you can say:

$$\log_ab\log_cd=\frac{\ln b}{\ln a}\frac{\ln d}{\ln c}=\frac{\ln e^{\ln b \ln d}}{\ln e^{\ln a \ln c}}=\log_{e^{\ln a \ln c}}{e^{\ln b \ln d}}$$

Etc.

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For reference:

I've found the following to work, but it seems like a bit of a hack and doesn't really help computation:

$$\log_a(b)*\log_c(d)=\log_\sqrt[\log_c(d)]{a}(b)$$ $$\log_a(b)/\log_c(d)=\log_a^{\log_c(d)}(b)$$

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