I want to express this into a single logarithm without using the calculator.
$$2\log_2 3 \cdot \log_3 2$$
My calculator's log function has only log base 10. It's easy to change the base to 10 and do it but I want to express this into a single logarithm without a calculator. How am I suppose to change their bases to be the same to apply further log rules?
$\endgroup$4 Answers
$\begingroup$We have that $2\log_2(3)\log_3(2)$. But is equal to $\log_3\left( 2^{2\log_2(3)} \right)=\log_3(3^2)=2$
$\endgroup$ $\begingroup$Hint: $$\log_a(b)=\frac{1}{\log_b(a)}.$$
$\endgroup$ $\begingroup$You want to find the value of $2xy$ where $2^x=3$ and $3^y=2$.
Raise the first of these equations to the $y^{th}$ power to get $xy$ in the exponent $$(2^x)^y=2^{xy}=3^y=2$$
Hence $xy=1$.
I sometimes find this kind of way through simpler than keeping track of the logs.
$\endgroup$ 1 $\begingroup$Also, $$2\log_23\log_32=2\cdot\frac{\ln3}{\ln2}\cdot\frac{\ln2}{\ln3}=2$$
$\endgroup$ 1
