Задача 77116 ...

662ac29927235c5f4601b706

4/25/2024 8:53:30 PM

Докажите числовое равенство
log_4(4√6-10)^2 + log_8(4√6+10)^3 = 2.

5f3ea7e3faf909182968ddd9

4/28/2024 11:38:24 AM

★

[m]log_{4}(4\sqrt{6}-10)^2=log_{2^2}(4\sqrt{6}-10)^2=\frac{2}{2}\cdot log_{2}|4\sqrt{6}-10|=log_{2}(10-4\sqrt{6})[/m]

[m]4\sqrt{6}<10[/m], так как [m](4\sqrt{6})^2<10^2[/m]

[m]log_{8}(4\sqrt{6}+10)^3=log_{2^3}(4\sqrt{6}+10)^3=\frac{3}{3}\cdot log_{2}(4\sqrt{6}+10)[/m]



[m]log_{4}(4\sqrt{6}-10)^2+log_{8}(4\sqrt{6}+10)^3=log_{2}(10-4\sqrt{6})+ log_{2}(4\sqrt{6}+10)= log_{2}(10-4\sqrt{6})\cdot log_{2}(4\sqrt{6}+10) =[/m]

[m]=log_{2}(10^2-(4\sqrt{6})^2)=log_{2}(100-96)=log_{2}4=2[/m]