Question

Daca log in baza 2 din 3 = a,calculati log in baza 12 din 18,in functie de a.​

Answer 1

$a = \log_{2}3 = \dfrac{\ln 3}{\ln 2} \\ \\\\ \log_{12}18 = \dfrac{\ln 18}{\ln 12} = \dfrac{\ln(9\cdot 2)}{\ln(3\cdot 4)}=\\ \\ =\dfrac{\ln 9+\ln 2}{\ln 3+\ln 4}= \dfrac{\ln 3^2 + \ln 2}{\ln 3 + \ln 2^2 }=\\ \\ =\dfrac{2\ln 3+\ln 2}{\ln 3+2\ln 2}^{(\,\ln 2} = \dfrac{\frac{2\ln 3}{\ln 2}+\frac{\ln 2}{\ln 2}}{\frac{\ln 3}{\ln 2}+\frac{2\ln 2}{\ln 2}} =\boxed{\dfrac{2a+1}{a+2}}$

Answer 2

$\log_{12}18=\frac{\log_2 18}{\log_2 12}=\\ \\ =\frac{\log_2 (9\cdot 2)}{\log_2 (3\cdot 4)}=\\ \\ =\frac{\log_2 9+\log_2 2}{\log_2 3+\log_2 4}=\\ \\ =\frac{2\cdot \log_2 3+1}{\log_2 3+2}=\\ \\ =\frac{2a+1}{a+2}$