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$\log_{12}{18}\times\log_{24}{54} + 5(\log_{12}{18}-\log_{24}{54})=1$ I know this isn't a difficult task but it's just killing me. I have tried many things, among which was base transformation to 12 and expressing every logarithm in terms of $\log_{12}{3}$ and $\log_{12}{2}$ but every time I try to do it, I mess up something.

2010年11月23日 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

$\begingroup$ Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level.

2015年8月26日 · The question: $$\log_9 (a) + \log_{12} (b) = \log_{16} (a+b)$$ solve for $a/b$. It gives hints: put it all in terms of x.

I first simplified it as $$\log_{12}18=\frac{\log_{9}18}{\log_{9}12}$$ and then proved that the numerator $\log_918$ is irrational by simplifying it to $1 + \log_9 2$ and proving $\log_9 2$ is irrational by contradiction: let $\log_9 2$ be rational so: $$\log_9 2 = \frac{m}{n}$$ $$9^m=2^n$$ which is a contradiction because it is saying that one ...

If $\log x+\log(2x-5)=\log 96–\log 8$, then $\log (2x^2-5x)=\log 12$ so that $2x^2-5x=12$. Write this as ...

Using the base-change formula, we can write this as: $\large\frac{log 18}{log 12} = a \\[0.4cm] \large\frac{2log 3+log 2}{log 3+2log 2}=a\\[0.4cm] 2 log 3 + log 2=a ...

One can write $$ 2^{2\log_2 3} = \left(2^{\log_2 3}\right)^2 = 3^2 = 9,\tag1 $$ so that is rational. But in doing that you don't need to know anything at all about rational or irrational numbers until that final step where you observe that $9$ is rational.

$2^{10} \approx 10^3$, and taking 120th roots, $2^{1/12} \approx 10^{1/40}$. and a "musical" fact: many rational numbers with small numerator and denominator can be approximated as powers of $2^{1/12}$. I call this a "musical" fact because $2^{1/12}$ is the frequency ratio corresponding to an (equal-tempered) semitone.