500=200*103^2x eching

$$500 = 200 \times 103 ^ { 2 x }$$

x uchun yechish

$$x=\frac{\log_{103}\left(\frac{5}{2}\right)}{2}\approx 0,098850519$$

Logaritm ifodasidan foydalanish qadamlari

Ikki tarafini 200 ga bo‘ling.

$$\frac{500}{200}=103^{2x}$$

\frac{500}{200} ulushini 100 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$\frac{5}{2}=103^{2x}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$103^{2x}=\frac{5}{2}$$

Tenglamaning ikkala tarafiga tegishli logaritmni chiqarish.

$$\log(103^{2x})=\log(\frac{5}{2})$$

Darajaga ko'tarigan logaritm raqami raqam logaritmining darajasidir.

$$2x\log(103)=\log(\frac{5}{2})$$

Ikki tarafini \log(103) ga bo‘ling.

$$2x=\frac{\log(\frac{5}{2})}{\log(103)}$$

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

$$2x=\log_{103}\left(\frac{5}{2}\right)$$

Ikki tarafini 2 ga bo‘ling.

$$x=\frac{\ln(\frac{5}{2})}{2\ln(103)}$$

x uchun yechish (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(103)}+\frac{\log_{103}\left(\frac{5}{2}\right)}{2}$$

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