3.
a) (lgx)²-2*lgx=3
lgx=t
t²-2x-3=0 D-16
t₁=3 lgx=3 x₁=10³=1000
t₂=-1 lgx=-1 x₂=10⁻¹=0,1.
b) 2⁶ˣ⁻¹-7*2³ˣ⁻¹=4
(1/2)*2⁶ˣ-(7/2)*2³ˣ-4=0 |×2
2⁶ˣ-7*2³ˣ=8=0
2³ˣ⁻¹=t>0
t²-7t-4=0 D=81
t₁=8 2³ˣ=8 2³ˣ=2³ 3x=3 x=1.
t₂=-1∉
Ответ:x=1.
c) log₃x-logₓ9=1
log₃x-1/log₃²x-1=0
log³x-1/(1/2)log₃x-1=0
(1/2)*log₃²x-1-(-1/2)*log₃ˣ=0 |×2
log₃²x-log₃x-2=0
log₃x=t
t²-t-2=0 D=9
t₁=2 log₃x=2 x=3² x₁=9
t₂=-1 log₃x=-1 x=3⁻¹ x₂=1/3.
d) log₄(5x-4)-3/((log₄(5x-4)-1)+1=0
5x-4=z
log₄z-3/(log₄z-1)+1=0
log₄²z-log₄z-3+log₄z-1=0
log₄²z-4=0
(log₄z-2)(log₄z+2)=0
log₄z=2 z=4²=16 5x-4=16 5x=20 x₁=4
log₄z=-2 z=4⁻²=1/16 5x-4=1/16 5x=4¹/₁₆=65/16 x₂=13/16.
e) 4ˣ-6*4⁻ˣ=2,5
4ˣ-6/4ˣ-2,5=0
4²ˣ-2,5*4ˣ-6=0
4ˣ=t>0
t²-2,5t-6=0 D=30,25
t₁=4 4ˣ=4¹ x=1
t₂=-1,5∉
Ответ:x=1