Question

Oblicz.

1.

a) log₂ 0,125

b) log₄ 2

c) log₄ 8

d) log₄ \frac{1}{1024}

e) log₄ \frac{1}{\sqrt{5}}

f) log\sqrt{2} 4

g) log\sqrt{2} 32

h) log₃ 3

2.

a) log₀,₂ 125

b) log⅙ \frac{1}{216}

c) log⅓ √3

d) log½ 2√2

3.

a) log 100+log₄ \frac{1}{16}

b) log 10⁷-log₀,₅ 8

c) log \sqrt[4]{10} - log₃ \sqrt[4]{3}

Answer 1

Definicja logarytmu:

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zad 1

a) log₂ 0,125=c <=> 2^c=0,125

2^c=0,125

2^c=1/8

2^c=8⁻¹

2^c=2⁻³

c=-3

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b) log₄ 2=c <=> 4^c=2

4^c=2

2^{2c}=2

2c=2

c=1/2

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c) log₄ 8=c <=> 4^c=8

4^c=8

2^{2c}=2³

2c=3

c=3/2

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d) log₄ 1/1024 = c <=> 4^c=1024

4^c=1024

2^{2c}=2^10

2c=10

c=5

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e) log₄ 1/√2 = c <=> 4^c=1/√2

4^c=1/√2

2^{2c}=√2⁻1

2^{2c}=2⁻²

2c=-2

c=-1

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f) log\sqrt{2} 4=c <=> √2^c=4

√2^c=4

2^{2c}=2²

2c=2

c=1

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g) log\sqrt{2} 32=c <=> √2^c=32

√2^c=32

2^{2c}=2⁵

2c=5

c=5/2

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h) log₃ 3=c <=> 3^c=3

3^c=3¹

c=1

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zad 2

a) log₀,₂ 125=c <=> (1/5)^c=125

(1/5)^c=125

5^{-c}=5³

c=-3

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b) log⅙ 1/216 =c <=> (1/6)^c=1/216

(1/6)^c=1/216

(1/6)^c=(1/6)³

c=3

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c) log⅓ √3 =c <=> (1/3)^c=√3

(1/3)^c=√3

3^{-c}=3²

c=-2

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d) log½ 2√2 =c <=> (1/2)^c=2√2

(1/2)^c=2√2

2^{-c}=2¹*2^{1/2}

2^{-c}=2^{3/2}

c=-3/2

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zad 3

a) log 100+log₄ 1/16=2+(-2)=0

 log 100=c <=> 10^c=100

10^c=100

10^c=10²

c=2

log₄ 1/16 =c <=> 4^c=1/16

4^c=1/16

4^c=4^{-2}

c=-2

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b) log 10⁷-log₀,₅ 8=7-(-3)=10

 log 10⁷=c <=> 10^c=10⁷

10^c=10⁷

c=7

log₀,₅ 8=c <=> (1/2)^c=8

(1/2)^c=8

2^{-c}=2³

c=-3

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c) log ⁴√10 - log₃ ⁴√3=1/4 - 1/4 =0

log ⁴√10=c <=> 10^c=⁴√10

10^c=⁴√10

10^c=10^{1/4}

c=1/4

log₃ ⁴√3=c <=> 3^c=⁴√3

3^c=⁴√3

3^c=3^{1/4}

c=1/4