Question

How do you use the properties of logarithms to rewrite and simplify the logarithmic expression of $ {\log _2}\left( {{7^2}{{.4}^7}} \right) $ ?

Answer 1

Hint: Since the logarithms are involved, we will have to use the properties of logarithms. You will have to mention the property used while starting the solution one by one and then continue with the simplification. The base will not matter while using the properties of logarithms.

Complete step-by-step answer:
We are given a logarithmic expression and told to simplify the expression.
Obviously we will have to use the logarithmic rules and follow them to simplify the expression given.
First we will have to deal with the multiplication.
So the rule goes like
 $ {\log _x}a.b = {\log _x}a + {\log _x}b $
Applying the above property to our expression we get
  $ {\log _2}\left( {{7^2}{{.4}^7}} \right) = {\log _2}{7^2} + {\log _2}{4^7} $
Now we have individual terms which are added.
Also we know that $ {\log _x}{a^b} = b{\log _x}a $
We can use this property to the individual terms in our expression. So we get
 $ {\log _2}\left( {{7^2}{{.4}^7}} \right) = 2{\log _2}7 + 7{\log _2}4 $
We can further write it as
 $ {\log _2}\left( {{7^2}{{.4}^7}} \right) = 2{\log _2}7 + 7{\log _2}{2^2} $
Solving further using the same property we get
\[
  {\log _2}\left( {{7^2}{{.4}^7}} \right) = 2{\log _2}7 + 7 \times 2{\log _2}2 \\
   \Rightarrow {\log _2}\left( {{7^2}{{.4}^7}} \right) = 2{\log _2}7 + 14{\log _2}2 \;
 \]
Hence this is the final simplified solution for the given expression.
So, the correct answer is “ $ \Rightarrow {\log _2}\left( {{7^2}{{.4}^7}} \right) = 2{\log _2}7 + 14{\log _2}2 $ ”.

Note: The properties of log are available only for multiplication, division and anything related to powers and exponents. There are no properties as such to simplify the problem which deal with addition and subtraction. Logarithms are usually used to simplify the large complicated problems.