Основное логарифмическое тождество:a в степени логарифм числа b по основанию a =b, т.е. a^(loga(b))=b
И дальше везде после log число или буква означает основание логарифма
1) 3^(log15(10))*5^(log15(10))
Читаем: логарифм 10 по основанию 15.
log15(10)=log3(10)/log3(15);
log15(10)=log5(10)/log5(15);
3^(log3(10)/log3(15))*5^(log5(10)/log5(15))=(3^(log3(10))^(1/log3(15))*(5^(log5(10))^(1/log5(15))=(10)^(1/log3(15))*(10))^(1/log5(15))=10^(1/(log3(15))+1/(log5(15))==10^(log15(3)+log15(5))=10^(log15(15))=10^1=10
2) 3^(log36(16))*2^(log36(16))
Читаем: логарифм 16 по основанию 36.
log36(16)=log3(16)/log3(36);
log36(16)=log2(16)/log2(36);
3^(log3(16)/log3(36))*2^(log2(16)/log2(36))=(3^(log3(16))^(1/log3(36))*(2^(log2(16))^(1/log2(36))=(16)^(1/log3(36))*(16))^(1/log2(36))=16^(1/(log3(36))+1/(log2(36))=16^(log36(3)+log36(2))=16^(log36(6))=16^(1/2)=√16=4
3) 3^(2*log3(4)): 2^(0,5*log2(16)=( 3^(log3(4))^2: (2^(log2(16))^(0,5)=4^2:(16)^(0,5)=16:√16=16:4=4
4) (0,2)^(-1+log5(0,2))= (1/5)^(-1+log5(0,2))= (5^(-1))^(-1+log5(0,2))=(5)^(1-log5(0,2))=5^1/(5^(log5(1/5))=5:(1/5)=25