Odpowiedź:
2.111
a)
log(√5)5∛5 = x - w nawiasie przedstawiona podstawa logarytmu
(√5)ˣ = 5∛5
(5¹⁾²)ˣ = 5 * 5¹⁾³
5¹⁾²ˣ = 5¹⁺¹⁾³
1/2x = 1 1/3
x = 1 1/3 : 1/2 = 4/3 * 2 = 8/3 = 2 2/3
b)
log(∛3)27 = x
(∛3)ˣ = 27
(3¹⁾³)ˣ = 3³
3¹⁾³x = 3³
1/3x = 3
x = 3 : 1/3 = 3 * 3 = 9
c)
log₂8√2 = x
2ˣ = 8√2
2ˣ = 2³ * 2¹⁾² = 2³ ¹⁾² = 2⁷⁾²
x = 7/2 = 3 1/2
d)
log₁₎₃81√3 = x
(1/3)ˣ = 3⁴ * 3¹⁾² = 3⁴ ¹⁾² = 3⁹⁾² = (1/3)⁻⁹⁾²
x = - 9/2 = - 4 1/2
e)
log₄8⁴√2 = x
4ˣ = 8 * ⁴√2 = 2³ * 2¹⁾⁴ = 2³ ¹⁾⁴ = 2¹⁴⁾⁴
(2²)ˣ = 2¹⁴⁾⁴
2²ˣ = 2¹⁴⁾⁴
2x = 14/4
x = 14/4 : 2 = 14/4 * 1/2 = 14/8 = 1 6/8 = 1 3/4
f)
log₁₎₅25√5 = x
(1/5)ˣ = 25√5 = 5² * 5¹⁾² = 5² ¹⁾² = 5⁵⁾² = (1/5)⁻⁵⁾²
x = - 5/2 = - 2 1/2
g)
log(√3)∛9 = x
(√3)ˣ = ∛9
(3¹⁾²)ˣ = 9¹⁾³ = (3²)¹⁾³ = 3²⁾³
3¹⁾²ˣ = 3²⁾³
1/2x = 2/3
x = 2/3 : 1/2 = 2/3 * 2 = 4/3 = 1 1/3
h)
log(2√2)4√8 = x
(2√2)ˣ = 4√8
(2 * 2¹⁾²)ˣ = 2² * 8¹⁾² = 2² * (2³)¹⁾² = 2² * 2³⁾² = 2²⁺³⁾² = 2² ³⁾² = 2⁷⁾²
(2¹⁺¹⁾²)ˣ = 2⁷⁾²
(2¹ ¹⁾²)ˣ = 2⁷⁾²
(2³⁾²)ˣ = 2⁷⁾²
3/2x = 7/2
x = 7/2 : 3/2 = 7/2 * 2/3 = 7/3 = 2 1/3
Szczegółowe wyjaśnienie:
¹⁾² - znaczy potęga 1/2
² ³⁾² - znaczy w potędze 2 i 3/2 = 7/2
