Leysa 5^286=5^x

$$5 ^ { 286 } = 5 ^ { x }$$

Leystu fyrir x

$$x=286$$

Skref með skilgreiningu logra

Reiknaðu 5 í 286. veldi og fáðu 80430587335437951845921127710495140134505930956790981674787620735993532493360592592242243732067646706109375636523120697559743178513198594503598322878201257201714879929710377837182022631168365478515625.

$$80430587335437951845921127710495140134505930956790981674787620735993532493360592592242243732067646706109375636523120697559743178513198594503598322878201257201714879929710377837182022631168365478515625=5^{x}$$

Skipta um hliðar svo allir liðir breytunnar séu vinstra megin.

$$5^{x}=80430587335437951845921127710495140134505930956790981674787620735993532493360592592242243732067646706109375636523120697559743178513198594503598322878201257201714879929710377837182022631168365478515625$$

Taka logra beggja hliða jöfnunnar.

$$\log(5^{x})=\log(80430587335437951845921127710495140134505930956790981674787620735993532493360592592242243732067646706109375636523120697559743178513198594503598322878201257201714879929710377837182022631168365478515625)$$

Logri tölu hækkaður í veldi er veldi sinnum logra tölunnar.

$$x\log(5)=\log(80430587335437951845921127710495140134505930956790981674787620735993532493360592592242243732067646706109375636523120697559743178513198594503598322878201257201714879929710377837182022631168365478515625)$$

Deildu báðum hliðum með \log(5).

$$x=\frac{\log(80430587335437951845921127710495140134505930956790981674787620735993532493360592592242243732067646706109375636523120697559743178513198594503598322878201257201714879929710377837182022631168365478515625)}{\log(5)}$$

Af „change-of-base“ formúlunni\frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$x=\log_{5}\left(80430587335437951845921127710495140134505930956790981674787620735993532493360592592242243732067646706109375636523120697559743178513198594503598322878201257201714879929710377837182022631168365478515625\right)$$

Leystu fyrir x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(5)}+286$$

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