Réitigh (100)(102^2x)=500000

$$(100)( { 102 }^{ 2x } )=500000$$

Réitigh do x.

$$x=\frac{\log_{102}\left(5000\right)}{2}\approx 0.920783055$$

Céimeanna ag Baint Úsáid as Sainiú an Logartaim

Úsáid rialacha na n-easpónant agus na logartam chun an chothromóid a réiteach.

$$100\times 102^{2x}=500000$$

Roinn an dá thaobh faoi 100.

$$102^{2x}=5000$$

Ghlac logartam an dá thaobh den chothromóid.

$$\log(102^{2x})=\log(5000)$$

Is ionann logartam uimhreacha a ardaítear go cumhacht agus an chumhacht méadaithe faoi logartam na huimhreach.

$$2x\log(102)=\log(5000)$$

Roinn an dá thaobh faoi \log(102).

$$2x=\frac{\log(5000)}{\log(102)}$$

Leis an bhfoirmle athrú boinn \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$2x=\log_{102}\left(5000\right)$$

Roinn an dá thaobh faoi 2.

$$x=\frac{\log_{102}\left(5000\right)}{2}$$

Réitigh do x. (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(102)}+\frac{\log_{102}\left(5000\right)}{2}$$

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