# Simplification of different base logarithms

I'm in doubt on simplifying the expression: $\log_2 6 - \log_4 9$

Working on it I've got: $\log_2 6 - \dfrac{\log_2 9}{2}$

There's anyway to simplify it more ? I'm learning logarithms now so I'm not aware of all properties and tricks.

Thanks in advance

## 3 Answers

You can do a bit more simplification. The important properties of the log function are, for any base $a>0$,

• $\log_a(bc)=\log_ab+\log_ac$, so, for example, $\log_26=\log_22+\log_23$
• $\log_a(b/c)=\log_ab-\log_ac$
• $\log_ab^n=n\log_ab$
• $\log_ab=1/\log_ba$
• $(\log_ab)(\log_bc)=\log_ac$
• $\log_aa=1$

So, for example, we can simplify $\log_26-\log_49$ as \begin{align} \log_26-\log_49&=\log_2(2\cdot3)-\log_4(3^2) \\ &= \log_22+\log_23-2\log_43 &\text{using the first and third identities}\\ &=1+\log_23-2\log_43 &\text{using the sixth identity}\\ &=1+\log_23-2\log_42\log_23 &\text{using the fifth}\\ &=1+\log_23-2(\log_23)/\log_24 &\text{using the fourth}\\ &=1+\log_23-2(\log_23)/2 &\text{using the third}\\ &=1+\log_23-\log_23 &\text{using a bit of algebra}\\ &=1 \end{align}

• Thank you, this table will helps me a lot. but in the Latex representation, the base is being shadowed by the 'G' of log. – aajjbb Nov 18 '12 at 21:10
• @aajbb Yeah, that happens to a lot of people, myself included. The consensus is that this site's LaTeX processor doesn't play nicely with some browsers. Often, if you simply tell the browser to refresh the page, the problem with overlapping text goes away. – Rick Decker Nov 18 '12 at 21:16

You're correct so far. Bring out the factor of $\frac{1}{2}$ to get $\frac{1}{2}(2\log_{2}(6)-\log_{2}(9))$. Since $a\log(b)=\log(b^{a})$ $\log(a)-\log(b)=\log(\frac{a}{b})$, you get $2\log_{2}(6)=\log_{2}(6^{2})$, and $$\frac{1}{2}(\log_{2}(36)-\log_{2}(9))=\frac{1}{2}\left(\log_{2}\left(\frac{36}{9}\right)\right)=\log_{2}(4)/2=2/2=1$$

$log_26-\frac{log_29}{2}=log_22+log_23-1/2×（2×log_23）=1$