Réitigh (100)(1.02^2x)=500000

$$(100)( { 1.02 }^{ 2x } )=500000$$

Réitigh do x.

$$x = \frac{\log_{1.02} {(5000)}}{2} \approx 215.052100512$$

Céimeanna ag Baint Úsáid as Sainiú an Logartaim

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

$$100\times 1.02^{2x}=500000$$

Roinn an dá thaobh faoi 100.

$$1.02^{2x}=5000$$

Ghlac logartam an dá thaobh den chothromóid.

$$\log(1.02^{2x})=\log(5000)$$

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

$$2x\log(1.02)=\log(5000)$$

Roinn an dá thaobh faoi \log(1.02).

$$2x=\frac{\log(5000)}{\log(1.02)}$$

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

$$2x=\log_{1.02}\left(5000\right)$$

Roinn an dá thaobh faoi 2.

$$x=\frac{\ln(5000)}{2\ln(\frac{51}{50})}$$

Réitigh do x. (complex solution)

$$x=\frac{i\pi n_{1}}{\ln(1.02)}+\frac{\log_{1.02}\left(5000\right)}{2}$$

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