Réitigh 3.2^3x-2-48=0

$$3.2 ^ { 3 x - 2 } - 48 = 0$$

Réitigh do x.

$$x=-\frac{\log_{3.2}\left(\frac{25}{12288}\right)}{3}\approx 1.776067379$$

Céimeanna ag Baint Úsáid as Sainiú an Logartaim

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

$$3.2^{3x-2}-48=0$$

Cuir 48 leis an dá thaobh den chothromóid.

$$3.2^{3x-2}=48$$

Ghlac logartam an dá thaobh den chothromóid.

$$\log(3.2^{3x-2})=\log(48)$$

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

$$\left(3x-2\right)\log(3.2)=\log(48)$$

Roinn an dá thaobh faoi \log(3.2).

$$3x-2=\frac{\log(48)}{\log(3.2)}$$

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

$$3x-2=\log_{3.2}\left(48\right)$$

Cuir 2 leis an dá thaobh den chothromóid.

$$3x=\frac{\ln(48)}{\ln(\frac{16}{5})}-\left(-2\right)$$

Roinn an dá thaobh faoi 3.

$$x=\frac{\frac{\ln(48)}{\ln(\frac{16}{5})}+2}{3}$$

Réitigh do x. (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{3\ln(3.2)}-\frac{\log_{3.2}\left(\frac{25}{12288}\right)}{3}$$

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