9^n*3^5*27^3/2*21^4=27 eching

$$\frac { 9 ^ { n } \times 3 ^ { 5 } \times 27 ^ { 3 } } { 2 \times 21 ^ { 4 } } = 27$$

n uchun yechish

$$n=\frac{\log_{3}\left(4802\right)-7}{2}\approx 0,357952375$$

Logaritm ifodasidan foydalanish qadamlari

5 daraja ko‘rsatkichini 3 ga hisoblang va 243 ni qiymatni oling.

$$\frac{9^{n}\times 243\times 27^{3}}{2\times 21^{4}}=27$$

3 daraja ko‘rsatkichini 27 ga hisoblang va 19683 ni qiymatni oling.

$$\frac{9^{n}\times 243\times 19683}{2\times 21^{4}}=27$$

4782969 hosil qilish uchun 243 va 19683 ni ko'paytirish.

$$\frac{9^{n}\times 4782969}{2\times 21^{4}}=27$$

4 daraja ko‘rsatkichini 21 ga hisoblang va 194481 ni qiymatni oling.

$$\frac{9^{n}\times 4782969}{2\times 194481}=27$$

388962 hosil qilish uchun 2 va 194481 ni ko'paytirish.

$$\frac{9^{n}\times 4782969}{388962}=27$$

9^{n}\times \frac{59049}{4802} ni olish uchun 9^{n}\times 4782969 ni 388962 ga bo‘ling.

$$9^{n}\times \frac{59049}{4802}=27$$

Ikki tarafini \frac{4802}{59049} va teskari kasri \frac{59049}{4802} ga ko‘paytiring.

$$9^{n}=27\times \frac{4802}{59049}$$

\frac{4802}{2187} hosil qilish uchun 27 va \frac{4802}{59049} ni ko'paytirish.

$$9^{n}=\frac{4802}{2187}$$

Tenglamaning ikkala tarafiga tegishli logaritmni chiqarish.

$$\log(9^{n})=\log(\frac{4802}{2187})$$

Darajaga ko'tarigan logaritm raqami raqam logaritmining darajasidir.

$$n\log(9)=\log(\frac{4802}{2187})$$

Ikki tarafini \log(9) ga bo‘ling.

$$n=\frac{\log(\frac{4802}{2187})}{\log(9)}$$

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

$$n=\log_{9}\left(\frac{4802}{2187}\right)$$

n uchun yechish (complex solution)

$$n=\frac{\pi n_{1}i}{\ln(3)}+\frac{\log_{3}\left(4802\right)}{2}-\frac{7}{2}$$

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