23^2x+1=54 eching

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

x uchun yechish

$$x=\frac{\log_{23}\left(54\right)-1}{2}\approx 0,136101324$$

Logaritm ifodasidan foydalanish qadamlari

Tenglamani yechish uchun eksponent va logaritmlarning qoidalaridan foydalanish.

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

Tenglamaning ikkala tarafiga tegishli logaritmni chiqarish.

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

Darajaga ko'tarigan logaritm raqami raqam logaritmining darajasidir.

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

Ikki tarafini \log(23) ga bo‘ling.

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

Asosiy tenglamani almashtirish orqali \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

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

Tenglamaning ikkala tarafidan 1 ni ayirish.

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

Ikki tarafini 2 ga bo‘ling.

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

x uchun yechish (complex solution)

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

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