Réitigh 500=200*1.03^2x

$$500 = 200 \times 1.03 ^ { 2 x }$$

Réitigh do x.

$$x = \frac{\log_{1.03} {(2.5)}}{2} \approx 15.499456378$$

Céimeanna ag Baint Úsáid as Sainiú an Logartaim

Roinn an dá thaobh faoi 200.

$$\frac{500}{200}=1.03^{2x}$$

Laghdaigh an codán \frac{500}{200} chuig na téarmaí is ísle trí 100 a bhaint agus a chealú.

$$\frac{5}{2}=1.03^{2x}$$

Athraigh na taobhanna ionas go mbeidh na téarmaí inathraitheacha ar fad ar an taobh clé.

$$1.03^{2x}=\frac{5}{2}$$

Ghlac logartam an dá thaobh den chothromóid.

$$\log(1.03^{2x})=\log(\frac{5}{2})$$

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

$$2x\log(1.03)=\log(\frac{5}{2})$$

Roinn an dá thaobh faoi \log(1.03).

$$2x=\frac{\log(\frac{5}{2})}{\log(1.03)}$$

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

$$2x=\log_{1.03}\left(\frac{5}{2}\right)$$

Roinn an dá thaobh faoi 2.

$$x=\frac{\ln(\frac{5}{2})}{2\ln(\frac{103}{100})}$$

Réitigh do x. (complex solution)

$$x=\frac{i\pi n_{1}}{\ln(1.03)}+\frac{\log_{1.03}\left(2.5\right)}{2}$$

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