Question

Решите логарифмы. Три номера
$1) \frac{ lg27+lg3}{lg15-lg5} ...2) lg30 + lg20 - lg6 ...3) 0,01^{log_{10}3-1 }$

Answer 1

1) (lg27+lg3)/(lg15-log5)=lg(27*3)/lg(15/5)=lg81/lg3=lg3⁴/lg3=4*lg3/lg3=4.
2) lg30+lg20-lg6=lg(30*20/6)=lg(600/6)=lg100=lg10²=2*lg10=2.
3) 0,01^((log₁₀3)-1)=10^(-2*(lg3-1)=10^(lg3⁻²+2)=100*10^lg(1/9)=
=100*(1/9)=100/9.

Answer 2

$\mathtt{\frac{\lg27+\lg3}{\lg15-\lg5}=\frac{\lg(27*3)}{\lg(15:5)}=\frac{\lg81}{\lg3}=\log_381=4}$

$\mathtt{\lg30+\lg20-\lg6=\lg(\frac{30*20}{6})=\lg(5*20)=\lg100=2}$

$\mathtt{0,01^{\lg3-1}=(10^{-2})^{\lg3-1}=10^{2-2\lg3}=10^2*(10^{\lg9})^{-1}=\frac{100}{9}}$