[tex]1) \displaystyle \\
a)~~2^{70}-2^{69}-2^{68} = 2^{68}(2^2-2^1-1)=2^{68}(4-2-1)=2^{68} \times 1 = 2^{68} \\
b)~~7^n+7^{n+1}+7^{n+2}=7^n(1+7+7^2)=7^n(1+7+49)=57 \times 7^n \\
c)~~3^{n+1}\cdot 5^{n+2}-15^n=3^n \cdot 3 \cdot 5^n \cdot 5^2 + 15^n =15^n \cdot 3 \cdot 25-15^n = \\ = 15^n \cdot 75-15^n =15^n (75-1) =74 \cdot 15^n[/tex]
[tex]2) \displaystyle \\
a)~~ 3^{21} = 3^x ~~~\Longrightarrow ~~x = 21 \\
b)~~ x^2=81;~~~~x^2=9^2;~~~\Longrightarrow ~~x = 9 \\
c)~~ 7^x = 49;~~~~ 7^x = 7^2; ;~~~\Longrightarrow ~~x = 2 \\
d)~~ 2^{2x}=16^4;~~~~2^{2x}=(2^4)^4;~~~~2^{2x}=2^{4\times 4} \\
~~~~~~ 2^{2x}=2^{16};~~~~2x = 16;~~~~x= \frac{16}{2}=8 \\
e)~~ 5^x=125^2;~~~~5^x=(5^3)^2;~~~~5^x=5^{3 \times 2};~~~~x=3 \times 2 =6 \\
f)~~ 10^{x-3} = 100;~~~~10^{x-3} = 10^2;~~~~\Longrightarrow ~~x-3 =2;~~\Longrightarrow ~~x=5[/tex]
