Question

Find the value of:
i) $\log_{\frac{1}{2}} 8$
ii) $\log_{5} 0.008$

Answer 1

Answer:

i) -3

ii) -3

Step-by-step explanation:

Hi,

Let x = log₁₎₂8

Using the definition of logarithm, we can rewrite as

(1/2)ˣ = 8

Writing 8 = 2³, we get

(1/2)ˣ = 8 = 2³

But we can write 1/2 as 2⁻¹

So, (1/2)ˣ = (2⁻¹)ˣ

Using the property (xᵃ)ⁿ = xᵃⁿ,

we can write  (2⁻¹)ˣ = 2⁻ˣ

So, we get 2⁻ˣ = 2³

Since, exponents are equal, power should also be equal

Hence, x = -3.

ii) Let x = log₅0.008

Using the definition of logarithm, we can rewrite as

5ˣ = 0.008

We can write 8 as 1/125 = 5⁻³

Hence , 5³ = 5⁻ˣ

Since, bases are equal  exponents should be equal

Hence, x = -3

Hope, it helps !

Answer 2

Answer:

-3

-3

Step-by-step explanation:

********************************

We know the logarithmic laws:

$1 )log_{a}a = 1\\\\2) log_{a}n^{m} =mlog_{a}n\\\\_____________________\\1 ) log_{1/2}8 \\\\= log_{1/2} 2^{3} \\\\= log_{1/2}(1/2)^{-3} \\\\\\= -3 \\\\2 ) log_{5}(0.008) \\\\= log_{5}( 8/1000 ) \\\\= log_{5}(\frac{1}{125}) \\\\= log_{5}(5)^{-3} \\\\= -3$