Réitigh 3(5^2x-1)+14=92

$$3 ( 5 ^ { 2 x - 1 } ) + 14 = 92$$

Réitigh do x.

$$x = \frac{\log_{5} {(26)} + 1}{2} \approx 1.5121846$$

Céimeanna ag Baint Úsáid as Sainiú an Logartaim

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

$$3\times 5^{2x-1}+14=92$$

Bain 14 ón dá thaobh den chothromóid.

$$3\times 5^{2x-1}=78$$

Roinn an dá thaobh faoi 3.

$$5^{2x-1}=26$$

Ghlac logartam an dá thaobh den chothromóid.

$$\log(5^{2x-1})=\log(26)$$

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

$$\left(2x-1\right)\log(5)=\log(26)$$

Roinn an dá thaobh faoi \log(5).

$$2x-1=\frac{\log(26)}{\log(5)}$$

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

$$2x-1=\log_{5}\left(26\right)$$

Cuir 1 leis an dá thaobh den chothromóid.

$$2x=\log_{5}\left(26\right)-\left(-1\right)$$

Roinn an dá thaobh faoi 2.

$$x=\frac{\log_{5}\left(26\right)+1}{2}$$

Réitigh do x. (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(5)}+\frac{\log_{5}\left(130\right)}{2}$$

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