Is there an identity/simplification for multiplying or dividing logarithms with different bases and values?

Question

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!

Answer 1

For reference:

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

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

Answer 2

No, there is no such identity, unless you count something like

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

Answer 3

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.