MHB Logarithmic Equation solve log_(3x)3+log_(x/3)3=5/12

  • Thread starter Thread starter Yankel
  • Start date Start date
  • Tags Tags
    Logarithmic
Yankel
Messages
390
Reaction score
0
Dear all,

I wish to solve the following logarithmic equation:

\[log_{3x}3+log_{\frac{x}{3}}3=\frac{5}{12}\]

My intuition was to start with changing the base of both logarithms to 10 (or any other number), but couldn't continue from there. Can you assist please ? Is there a meaning to the fact that both bases involves 3 in them ?

Thank you in advance.
 
Mathematics news on Phys.org
Yes, you certainly want the two logarithms to the same base but there is nothing special about base 10. Since one logarithm is already to base "3x" I would be inclined to change the other to base 3x also. If y= log_{x/3}(3) then 3= (x/3)^y= x^y/3^y. So x^y= 3^{y+1} and then (3x)^y= 3^yx^y= 3^{2y+ 1}. Taking the logarithm, base 3x, of both sides, y= log_{3x}(3^{2y+1})= (2y+1)log_{3x}(3). Solving that for y, (1- 2log_{3x}(3))y= log_{3x}(3) so y= log_{x/3}(3)= \frac{log_{3x}(3)}{1- 2log_{3x}(3)}

The original equation, log_{3x}(3)+ log_{x/3}(3)= \frac{5}{12} becomes log_{3x}(3)\left(1+ \frac{log_{3x}(3)}{1-2log_{3x}(3)}\right)= \frac{5}{12}.

To simplify, let y= log_{3x}(3) so the equation is y\left(1+ \frac{y}{1- 2y}\right)= \frac{5}{12}. Solve that for y then solve log_{3x}(3)= y for x.
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
Back
Top