Whakaoti i te frac{5^{m}*5^3*5^-2}{5^-5}=5^12

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

Whakaoti mō m

$$m=6$$

Ngā Upane e Whakamahi ana i te Tautuhinga o te Taupū Kōaro

Hei whakarea i ngā pū o te pūtake kotahi, me tāpiri ō rātou taupū. Tāpiria te 3 me te -2 kia riro ai te 1.

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

Hei whakawehe i ngā pū o te pūtake kotahi, tangohia te taupū o te tauraro mai i te taupū o te taurunga.

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

Tātaihia te 5 mā te pū o 12, kia riro ko 244140625.

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

Tātaihia te 5 mā te pū o 6, kia riro ko 15625.

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

Whakawehea ngā taha e rua ki te 15625.

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

Whakawehea te 244140625 ki te 15625, kia riro ko 15625.

$$5^{m}=15625$$

Tuhia te tau taupū kōaro o ngā taha e rua o te whārite.

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

Ko te taupū kōaro o tētahi tau ka hīkina ki tētahi pū ko te pū whakarea ki te taupū kōaro o taua tau.

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

Whakawehea ngā taha e rua ki te \log(5).

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

Mā te tikanga tātai huri pūtake \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

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

Whakaoti mō m (complex solution)

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

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