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

Oct 13, 2017

$1 + 2 {\log}_{10} 2$

#### Explanation:

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

$\text{ } {\log}_{10} \left(25 \cdot 10\right)$

Using the multiplication property of logarithms (${\log}_{a} \left(x \cdot y\right) = {\log}_{a} x + {\log}_{a} y$); and writing 25 as ${5}^{2}$

$\text{ } {\log}_{10} \left({5}^{2} \cdot 10\right) = {\log}_{10} 10 + {\log}_{10} {5}^{2}$

Knowing that ${\log}_{a} a = 1$ (${\log}_{10} 10 = 1$), and the power property of logarithms (${\log}_{a} {x}^{n} = n {\log}_{a} x$)

$\text{ } {\log}_{10} 10 + {\log}_{10} {5}^{2} = 1 + 2 {\log}_{10} 5$

Hope that helped!!

Oct 13, 2017

Option 3) $3 - 2 {\log}_{10} 2$

#### Explanation:

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

${\log}_{a} \left({b}^{c}\right) = c \cdot {\log}_{a} b \textcolor{w h i t e}{a a a a a a} \text{Exponent Property}$

${\log}_{a} \left(\frac{b}{c}\right) = {\log}_{a} b - {\log}_{a} c \textcolor{w h i t e}{a a a a a a} \text{Quotient Property}$

${\log}_{a} a = 1 \textcolor{w h i t e}{a a a a a a a} \text{Identity Property}$

Begin by recognizing that $250 = \frac{1000}{4}$, and rewrite the original logarithm:

${\log}_{10} 250 = {\log}_{10} \left(\frac{1000}{4}\right)$

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

${\log}_{10} \left(\frac{1000}{4}\right) = {\log}_{10} 1000 - {\log}_{10} 4$

Rewrite both logarithms using exponents:

${\log}_{10} 1000 - {\log}_{10} 4 = {\log}_{10} \left({10}^{3}\right) - {\log}_{10} \left({2}^{2}\right)$

Apply the Exponent Property:

${\log}_{10} \left({10}^{3}\right) - {\log}_{10} \left({2}^{2}\right) = 3 \cdot {\log}_{10} 10 - 2 \cdot {\log}_{10} 2$

Finally, apply the Identity Property:

$3 \cdot {\log}_{10} 10 - 2 \cdot {\log}_{10} 2 = 3 \left(1\right) - 2 \cdot {\log}_{10} 2 = 3 - 2 {\log}_{10} 2$