Question

What is the correct option from given options? pls explain your answer

Answer 1

Answer nº 2

`1 + 2log_10 2`

Explanation:

Let's start by putting 250 in the form of 25*10:

`" " log_10 (25*10)`

Using the multiplication property of logarithms (`log_a (x*y) = log_a x + log_a y`); and writing 25 as `5^2`

`" " log_10 (5^2*10) = log_10 10 + log_10 5^2`

Knowing that `log_a a = 1` (`log_10 10 = 1`), and the power property of logarithms (`log_a x^n = n log_a x`)

`" "log_10 10 + log_10 5^2 = 1 + 2 log_10 5`

Hope that helped!!

Answer 2

Option 3) `3-2log_(10) 2`

Explanation:

There are a few ways to do this, but this is the method I used. It will use the following properties of logarithms:

`log_a (b^c) = c*log_a b color(white)(aaaaaa)"Exponent Property"`

`log_a (b/c) = log_a b - log_a c color(white)(aaaaaa)"Quotient Property"`

`log_a a = 1 color(white)(aaaaaaa)"Identity Property"`

Begin by recognizing that `250 = 1000/4`, and rewrite the original logarithm:

`log_10 250 = log_10 (1000/4)`

Now apply the Quotient Property above to split this logarithm into two separate logarithms:

`log_10 (1000/4) = log_10 1000 - log_10 4`

Rewrite both logarithms using exponents:

`log_10 1000 - log_10 4 = log_10 (10^3) - log_10 (2^2)`

Apply the Exponent Property:

`log_10 (10^3) - log_10 (2^2) = 3*log_10 10 - 2*log_10 2`

Finally, apply the Identity Property:

`3*log_10 10 - 2*log_10 2 = 3(1) - 2*log_10 2 = 3 - 2 log_10 2`