# How do I square a logarithm?

How do I square $\log_2(3)$. Does it become $2\log_2(3)$ ?

No. $(\log_2(3))^2$ can't be simplified.

However, $\log_2(3^2)=2\log_2(3)$.

• Well, one does have $(\log_2(3))^2=\log_2(3^{\log_2(3)})$, but I'm not sure whether you can call that a simplification. – Marc van Leeuwen May 7 '15 at 14:09

No, it doesn't. Logarithms follow this rule: $$\log_b (a^c) = c\log_b a,$$ while your statement says that $$(\log_b a)^c=c\log_b a,$$ which is basically saying the same as $x^y=yx$.

$\log_2(3) \approx 1.58496$ as you can easily verify.

$(\log_2(3))^2 \approx (1.58496)^2 \approx 2.51211$.

$2 \log_2(3) \approx 2 \cdot 1.58496 \approx 3.16992$.

$2^{\log_2(3)} = 3$.

Do any of those appear to be equal?

(Whenever you are wondering whether some general algebraic relationship holds, it's a good idea to first try some simple numerical examples to see if it is even possible.)

Actually, the only way that $(\log_2(3))^2 = 2 \log_2(3)$ could hold is if $\log_2(3)$ were equal to 2 or 0. That is clearly false since $2^2 = 4 \ne 3$ and $2^0 = 1 \ne 3$.

• I am an engineer, and I would say that 3rd and 4rt are equal, indeed. :P – Ander Biguri May 8 '15 at 11:56
• You: Actually, the only way that $(\log_2(3))^2 = 2 \log_2(3)$ could hold is if $\log_2(3)$ were equal to 2. Maybe that should be "2 or 0" (the solutions to $x^2=2x$). For example, $(\log_2(1))^2 = 2 \log_2(1)$. – Jeppe Stig Nielsen May 8 '15 at 14:48
• @JeppeStigNielsen: Thanks, fixed, – Nate Eldredge May 8 '15 at 14:48

No.The best you can do is $$(\log_2 3)^2=\log_2 3\cdot \log_2 3=\log_2 (3^{\log_2 3})$$

• log3 on base 2 cant be 2^ log 3 on base 2? – Raunak Banerjee May 7 '15 at 16:46

Agreeing with the rest of the answers here, you cannot simplify any further. You could, however, do a change of base with the logs and put them in base $10$. We have the formula $$\log_bx=\frac{\log_ax}{\log_ab}$$where $a$ can be any base you want. Most common base is $10$. So we have, $$(\log_23)^2=\left(\frac{\log_{10}3}{\log_{10}2}\right)^2=\left(\frac{\log3}{\log2}\right)^2=\frac{\log^23}{\log^22}$$

• For what purpose? – Michael Le Barbier Grünewald May 8 '15 at 11:26
• @MichaelGrünewald To make a calculator do it? – KSmarts May 8 '15 at 14:50
• @KSmarts In which world does this relate to the question asked? – Michael Le Barbier Grünewald May 8 '15 at 14:57

\begin{eqnarray} (\log_{2}(3))^{2} &=& (\log_{2}(3))(\log_{2}(3)) \\ &=& \log_{2}(3^{\log_{2}(3)}) \end{eqnarray} That's as simple as it gets!