Question

Вычислить:
1) $2^{frac{log_2 5}{log_5 2}}-5^{log_2 5}$
2) $3^{2+frac{log_3 4}{log_4 3}}-9cdot4^{frac1{log_43}}+4^{1+log_4 25}$

Answer 1

1)
2^(log(2)5 / log(5)2) - 5^log(2)5 =
= 2^(log(2)5 * log(2)5) - 5^log(2)5 =
= (2^log(2)5 )^log(2)5  - 5^log(2)5 =
= 5^log(2)5 - 5^log(2)5 = 0

2)
3^( 2 + (log(3)4)/(log(4)3)) - 9*4^(1 / log(4)3) +4^(1 + log(4)25) =
= 3^2 * (3^log(3)4)^log(3)4 - 9*4^log(3)4 +4*4^log(4)25 =
= 3^2 * 4^log(3)4 - 9 * 4^log(3)4 +4*25 =
= 9 * 4^log(3)4 - 9 * 4^log(3)4 +4*25 = 4*25 = 100


$2^{frac{log_25}{ log_52}} - 5^{log_25} = 2^{log_25 *log_25} - 5^{log_25} =$

$= (2^{log_25} )^{log_25} - 5^{log_25}= 5^{log_25} - 5^{log_25} = 0$


$3^{ 2 + frac{log_34}{log_43}} - 9*4^{frac{1 }{ log_43}} +4^{1 + log_425} =$

$= 3^2 * (3^{log_34})^{log_34} - 9*4^{log_34} +4*4^{log_425} =$

$= 3^2 * 4^{log_34} - 9 * 4^{log_34} +4*25 = \\= 9 * 4^{log_34} - 9 * 4^{log_34} +4*25 = 4*25 = 100$