# Is $\lg_{\frac{2}{3}}{\frac{1}{n}} = \lg_{1.5}{n}$

On 1st line is something like

The size of the problem at level k is (2/3)kn. The size at the lowest level is 1, so setting (2/3)kn = 1, the depth is k = log1.5 n (divide both sides by (2/3)k, take logs base 1.5).

the main question is I have

$$1 = \left(\frac{2}{3}\right)^k \cdot n$$

I want to find k. I did

$$\left(\frac{2}{3}\right)^k n=1$$ $$\left(\frac{2}{3}\right)^k=\frac{1}{n}$$ $$k\lg\frac{2}{3}=\lg\frac{1}{n}$$ $$k=\frac{\lg\frac{1}{n}}{\lg\frac{2}{3}}$$ $$k=\lg_{\frac{2}{3}}\frac{1}{n}$$

I suppose I could somehow convert it to the desired answer $\lg_{1.5}{n}$?

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For those interested in relatively unknown logarithm identities, see mathforum.org/kb/message.jspa?messageID=6470549 and my short note Harmonious logarithm identities, Mathematical Gazette 93 #526 (March 2009), 95-97. – Dave L. Renfro Nov 15 '11 at 15:54

Your calculations are correct. In order to convert it to the desired answer, you need to use that $\log a/b = \log a - \log b$. Then, you can write

$$k = \frac{\log 1/n}{\log 2/3} = \frac{- \log n}{-\log 3/2} = \log_{1.5}n.$$

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What they wanted you to do is

$$\left(\frac{2}{3}\right)^kn=1 \iff n=\left(\frac{3}{2}\right)^k$$

by dividing both sides by $\left(\frac{2}{3}\right)^k$. Then you can solve by hand again and get the desired result.

Alternatively you can use

$$\log_\frac{2}{3}(n^{-1})=-\log_\frac{2}{3}(n)=\log_\frac{3}{2}(n)$$

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Yes, you could. What you need is

$$\log\frac1x=-\log x\;.$$

You can use that to swap the fractions in both logarithms, and the resulting signs will cancel. However, perhaps slightly easier would be to divide the equation by $(2/3)^k$ instead of $n$ in the first place; then you'd immediately get $(3/2)^k=n$ and thus $k\log(3/2)=\log n$.

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