3(5^2x-1)+14=92 eching

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

x uchun yechish

$$x = \frac{\log_{5} {(26)} + 1}{2} \approx 1,5121846$$

Logaritm ifodasidan foydalanish qadamlari

Tenglamani yechish uchun eksponent va logaritmlarning qoidalaridan foydalanish.

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

Tenglamaning ikkala tarafidan 14 ni ayirish.

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

Ikki tarafini 3 ga bo‘ling.

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

Tenglamaning ikkala tarafiga tegishli logaritmni chiqarish.

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

Darajaga ko'tarigan logaritm raqami raqam logaritmining darajasidir.

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

Ikki tarafini \log(5) ga bo‘ling.

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

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

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

1 ni tenglamaning ikkala tarafiga qo'shish.

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

Ikki tarafini 2 ga bo‘ling.

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

x uchun yechish (complex solution)

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

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