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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?

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4 Answers 4

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

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

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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.

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  • $\begingroup$ Sorcery! Very nice. $\endgroup$
    – Peter Szilas
    Commented Nov 19, 2017 at 9:26
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Also, $$2\log_23\log_32=2\cdot\frac{\ln3}{\ln2}\cdot\frac{\ln2}{\ln3}=2$$

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  • $\begingroup$ Can down-voter explain us, why did you do it? $\endgroup$
    – Michael Rozenberg
    Commented Dec 11, 2019 at 4:34

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