expontialgleichung

Hier etwas ausführlicher:

5^(2x+7) = 3^(5x-1)          logarithmieren !

(2x+7)*log(5) = (5x-1)*log(3)        Klammern ausmultiplizieren  !

2x*log(5)+7*log(5) = 5x*log(3)-1*log(3)       ordnen !   x-er nach links !

2x*log(5)-5x*log(3) = -log(3)-7*log(5)            |  mal (-1)   und x ausklammern !

x(5*log(3)-2*log(5)) = log(3)+7*log(5)           durch die linke Klammer dividieren !

x =    $${\frac{\left({log}_{10}\left({\mathtt{3}}\right){\mathtt{\,\small\textbf+\,}}{\mathtt{7}}{\mathtt{\,\times\,}}{log}_{10}\left({\mathtt{5}}\right)\right)}{\left({\mathtt{5}}{\mathtt{\,\times\,}}{log}_{10}\left({\mathtt{3}}\right){\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{log}_{10}\left({\mathtt{5}}\right)\right)}}$$   = 5,436969....

Gruß radix !

Umformung:

 x =      $${\frac{\left({log}_{10}\left({\mathtt{3}}\right){\mathtt{\,\small\textbf+\,}}{\mathtt{7}}{\mathtt{\,\times\,}}{log}_{10}\left({\mathtt{5}}\right)\right)}{\left({\mathtt{5}}{\mathtt{\,\times\,}}{log}_{10}\left({\mathtt{3}}\right){\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{log}_{10}\left({\mathtt{5}}\right)\right)}}$$

Gruß radix !

@raidx Bitte noch den Rachenweg dazu, ich kann nicht nachvollziehen, wie du darauf kommst.