frac{5^{m}*5^3*5^-2}{5^-5}=5^12 eching

$$\frac { 5 ^ { m } \times 5 ^ { 3 } \times 5 ^ { - 2 } } { 5 ^ { - 5 } } = 5 ^ { 12 }$$

m uchun yechish

$$m=6$$

Logaritm ifodasidan foydalanish qadamlari

Ayni asosning daraja ko‘rsatkichlarini ko‘paytirish uchun ularning darajalarini qo‘shing. 3 va -2 ni qo‘shib, 1 ni oling.

$$\frac{5^{m}\times 5^{1}}{5^{-5}}=5^{12}$$

Ayni asosning daraja ko'rsatkichi bo'lish uchun maxrajning darajasini surat darajasidan bo'ling.

$$5^{6}\times 5^{m}=5^{12}$$

12 daraja ko‘rsatkichini 5 ga hisoblang va 244140625 ni qiymatni oling.

$$5^{6}\times 5^{m}=244140625$$

6 daraja ko‘rsatkichini 5 ga hisoblang va 15625 ni qiymatni oling.

$$15625\times 5^{m}=244140625$$

Ikki tarafini 15625 ga bo‘ling.

$$5^{m}=\frac{244140625}{15625}$$

15625 ni olish uchun 244140625 ni 15625 ga bo‘ling.

$$5^{m}=15625$$

Tenglamaning ikkala tarafiga tegishli logaritmni chiqarish.

$$\log(5^{m})=\log(15625)$$

Darajaga ko'tarigan logaritm raqami raqam logaritmining darajasidir.

$$m\log(5)=\log(15625)$$

Ikki tarafini \log(5) ga bo‘ling.

$$m=\frac{\log(15625)}{\log(5)}$$

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

$$m=\log_{5}\left(15625\right)$$

m uchun yechish (complex solution)

$$m=\frac{2\pi n_{1}i}{\ln(5)}+6$$

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