Math log proof

1. Apr 18, 2004

gnome

Would someone please show me why

$${ a^{log_cb} = b^{log_ca}$$

for all a, b and c.

2. Apr 18, 2004

deltabourne

I don't know if this is really a proof.. but just take the log of both sides:
$${ a^{log_cb} = b^{log_ca}$$
$$log_c { a^{log_cb} = log_c b^{log_ca}$$
$$({log_cb})({log_ca}) = ({log_ca})({log_cb})$$
using the property that:
$${log_ca^r} = r{log_ca}$$

3. Apr 18, 2004

matt grime

it is a proof. perhaps to make it appear more rigorous, you could write $$a^{log_cb}=c^{log_c(a^{log_cb})}$$ and similarly for the rhs, and say that the only way for c^xto be equal to c^y is if x=y.

4. Apr 18, 2004

gnome

Got it!

Thanks, folks.