Riješi (45*32){left(8/3right)}^2x=23

$$(45 \times 32) { \left( \frac{ 8 }{ 3 } \right) }^{ 2x } =23$$

Riješite za x

$$x=\frac{\log_{\frac{8}{3}}\left(\frac{23}{1440}\right)}{2}\approx -2,108880911$$

Koraci pomoću definicije logaritma

Koristite pravila eksponenata i logaritama za rješavanje jednačine.

$$1440\times \left(\frac{8}{3}\right)^{2x}=23$$

Podijelite obje strane s 1440.

$$\left(\frac{8}{3}\right)^{2x}=\frac{23}{1440}$$

Izračunajte logaritam obje strane jednačine.

$$\log(\left(\frac{8}{3}\right)^{2x})=\log(\frac{23}{1440})$$

Logaritam broja podignutog na stepen je stepen puta logaritam broja.

$$2x\log(\frac{8}{3})=\log(\frac{23}{1440})$$

Podijelite obje strane s \log(\frac{8}{3}).

$$2x=\frac{\log(\frac{23}{1440})}{\log(\frac{8}{3})}$$

Po formuli promjene osnove \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$2x=\log_{\frac{8}{3}}\left(\frac{23}{1440}\right)$$

Podijelite obje strane s 2.

$$x=\frac{\ln(\frac{23}{1440})}{2\ln(\frac{8}{3})}$$

Riješite za x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(\frac{8}{3})}+\frac{\log_{\frac{8}{3}}\left(\frac{23}{1440}\right)}{2}$$

$$n_{1}\in \mathrm{Z}$$