Réitigh 23^2x+1=54

$$23 ^ { 2 x + 1 } = 54$$

Réitigh do x.

$$x=\frac{\log_{23}\left(54\right)-1}{2}\approx 0.136101324$$

Céimeanna ag Baint Úsáid as Sainiú an Logartaim

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

$$23^{2x+1}=54$$

Ghlac logartam an dá thaobh den chothromóid.

$$\log(23^{2x+1})=\log(54)$$

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

$$\left(2x+1\right)\log(23)=\log(54)$$

Roinn an dá thaobh faoi \log(23).

$$2x+1=\frac{\log(54)}{\log(23)}$$

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

$$2x+1=\log_{23}\left(54\right)$$

Bain 1 ón dá thaobh den chothromóid.

$$2x=\log_{23}\left(54\right)-1$$

Roinn an dá thaobh faoi 2.

$$x=\frac{\log_{23}\left(54\right)-1}{2}$$

Réitigh do x. (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(23)}+\frac{\log_{23}\left(54\right)}{2}-\frac{1}{2}$$

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