$2\log_2 3 \cdot\log_3 2$ without using a calculator

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?

We have that $2\log_2(3)\log_3(2)$. But is equal to $\log_3\left( 2^{2\log_2(3)} \right)=\log_3(3^2)=2$

Hint: $$\log_a(b)=\frac{1}{\log_b(a)}.$$

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.

• Sorcery! Very nice. – Peter Szilas Nov 19 '17 at 9:26

Also, $$2\log_23\log_32=2\cdot\frac{\ln3}{\ln2}\cdot\frac{\ln2}{\ln3}=2$$

• Can down-voter explain us, why did you do it? – Michael Rozenberg Dec 11 '19 at 4:34