Հաշվել արտահայտության արժեքը
105
ա) log915 + log918 – log910 = log915 * 18/10 = log9 + 3 * 4 = log912
բ) log5 * (7 + 2<6) + log5 * (7 – 2<6) = log5 * (7 + 2 – <6) * (7 – 2 – <6) = log5 * (19 * -4 * 6) = log5 * (49 – 24) = log225 = 2
գ) 1/2log7 36 – log7 14 — 3log7 ∛21 = 1/2 log7 6² — log7 14 — log7 (∛21)³ = 1/2 · 2· log7 6 — log7 14 — log7 21 = log7 6 — log7 14 — log7 21 = log7 (6:14:21) = log7 (1/49) = log7 (7⁻²) = -2log7 7 = -2·1 = -2
դ) 2log1/56 – 1/2log1/5400 – 4log1/54<45 = log1/536 * 4/20 * 455 = log1/51/52 = 21/5
106
ա) lg(11 — 3x) < 2 — lg5
11 — 3x > 0, 3x < 11, x < 11/3
lg(11 — 3x) < 2lg(10) — lg5
lg(11 — 3x) < lg(100/5)
11 — 3x < 20
3x > — 9
x > — 3
Պատ՝ х ∈ (- 3; 11/3)
բ) lg(7x + 5) < 1 + lg3
lg(7x + 5) < lg(10) + lg(3)
lg(7x + 5) < lg(30)
7x < 30 — 5
7x < 25
x < 25/7,
x > -5/7
Պատ՝ х ∈ (-5/7; 25/7)
գ) log₂(4x — x²) < 5 + 2log₀,₅3
log₂(4x — x²) < log₂(2⁵ * 3‾²)
log₂(4x — x²) < log₂(32 * 1/9)
log₂(4x — x²) < log₂(32/9)
36x — 9x² — 32< 0
-9x² + 36x — 32 < 0
(3x — 8) * (3x — 4) > 0
х ∈ (8/3; ∞)
х ∈ (-∞; 4/3)
х ∈ (-∞; 4/3) U (8/3; +∞); х ∈ (0,4)
х ∈ (0; 4/3) U (8/3; 4)
դ) log₂(x² — 3x — 4) ≤ 2 + log√₂3
log₂(x² — 3x — 4) ≤ log₂ (2²) + log₂(3²)
log₂(x² — 3x — 4) ≤ log₂(36)
x² — 3x — 4 — 36 ≤ 0
x² + (x + 5) — 8(x+ 5) ≤ 0
(x + 5) * (x — 8) ≤ 0
{x ≤ -5; x ⪰ 8 {x ⪰-5; x ≤ 8
х ∈ ∅
х ∈ [-5, 8]
х ∈ [-5, 8], х ∈ (-∞; -1) U (4, +∞)
х ∈ [-5; -1) U (4; 8]
107
ա) log₄ (x + 3) ≤ log₄ (9x — 13)
log₄(x + 3) ≤ log₄(9x — 19), х ∈ (13/9; +∞)
x — 9x ≤ -13 — 3
-8x ≤ -16
x ≥ 2, x ∈ (13/9, +∞)
x∈ [2, +∞)
x ≥ 2