Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Prove that if $\alpha = \log_{12}18$ and $ \beta = \log_{24}54$ then $ \alpha \beta +5(\alpha - \beta)=1$

share|cite|improve this question
Do you know log properties, e.g., $\log_k(b)\log_b(x)=\log_k(x)$? – snarski Feb 18 '13 at 2:57
Is that an order? – copper.hat Feb 18 '13 at 2:57
@copper I assume you're asking the OP? If so, absolutely. chop chop. – snarski Feb 18 '13 at 2:58
@snarski: Indeed, the comment was aimed at the OP! – copper.hat Feb 18 '13 at 2:59
Sachin, where did you get this problem? – Will Jagy Feb 18 '13 at 20:05

Suppose we have positive integers $m,n$ with $m^2 + mn - n^2 = 1.$ This means they are consecutive fibonacci numbers $m = f_{2k}, \; \; n = f_{2k+1},$ for example $(m=1,n=1), \; \; (m=2,n=3), \; \; (m=5,n=8), \; \; (m=13,n=21). $ Next suppose we have real or complex variables $x,y.$ Next we take $$ \alpha = \frac{x + m y}{m x + y}, \; \; \; \beta = \frac{x + n y}{n x + y}. $$ Then, by putting on a common denominator, we can confirm that $$ \alpha \beta + (m+n)\alpha - (m+n) \beta = 1. $$ This is really very clever and not something I knew. I am wondering what other such things might be true, combining indefinite quadratic forms with rational functions in two variables that resemble linear fractional transformations.

As @lab has pointed out, the values of $x,y$ do not matter at all. However, for this problem they can be taken to be natural logarithms $x = \log 2, y = \log 3.$ Oh, for this problem $m=2,n=3.$

share|cite|improve this answer

$$\log_{12}18=\frac{\log_218}{\log_212}=\frac{\log_22+\log_23^2}{\log_22^2+\log_23}=\frac{1+2\log_23}{2+\log_23}$$ as $\log_ab=\frac{\log_cb}{\log_ca}$ where $a\ne1,b,c\ne1$ are positive real numers and $\log mn=\log m+\log n$

Similarly, simplify $\log_{24}54$ and equate the values of $\log_23$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.