How to calculate expression like this one below?
$\log_7(19) \log_2(5) = \log_x(y)$
Or at least where can I find any information about expressions like this. Examples and explanation that I found in google are for simple examples that I am able to calculate without help, but not for something like this one.
Remember that: $$\log_a(x)=\frac{\ln x}{\ln a}$$ So: $$\log_7(19)*\log_2(5)=\frac{\ln19\ln5}{\ln7\ln2}=\frac{\ln{5^{\ln{19}}}}{\ln{2^{\ln{7}}}}$$So the input is $5^{\ln19}$ and the base is $2^{\ln7}$. As you can see it is better to stick to that expression you had before.
One option:
$$\log_7{19}\cdot\log_25=\log_7\left(19^{\log_25}\right)$$
or
$$\log_7{19}\cdot\log_25=\log_2\left(5^{\log_7{19}}\right)$$
In general, you have:
$$\log_ab\cdot\log_cd=\log_c\left(d^{\log_ab}\right)$$
or
$$\log_ab\cdot\log_cd=\log_a\left(b^{\log_cd}\right).$$
This comes from the logarithm power rule:
$$\log_ab^p=p\log_ab$$
where $a,b>0\wedge a≠1\wedge p\in\mathbb R$.
