Question

Find $4\log 5+2\log 4$

Answer 1

Hint: We will use some familiar logarithmic identities to solve the given problem. We will use the logarithmic identity that says $n\log x=\log {{x}^{n}}.$ We will also use the logarithmic identity that says $\log x+\log y=\log xy.$ We have learnt the identity $\log {{10}^{n}}=n.$

Complete step by step answer:
Let us consider the given problem in which we are asked to find the value of $4\log 5+2\log 4.$
Here, as we can see, we need to find the sum where both the summands are logarithmic functions.
Now, we can write the given function as $4\log 5+2\log 4=2\times 2\log 5+2\log 4.$
Let us take $2$ out to get $2\left( 2\log 5+\log 4 \right).$
Let us consider the summands separately so that we can rewrite them in the feasible ways.
So, the first summand is $2\log 5.$
We have already learnt the logarithmic identity given by $n\log x=\log {{x}^{n}}.$
If we use this identity in the first summand with $n=2$ and $x=5,$ then we will get $2\log 5=\log {{5}^{2}}.$
From this, we will get $2\log 5=\log 25.$
Let us substitute for $2\log 5$ in the obtained form of the function.
We will get $2\left( 2\log 5+\log 4 \right)=2\left( \log 25+\log 4 \right).$
 We are also familiar with the logarithmic identity given by $\log x+\log y=\log xy.$
So, we can use this identity in the above obtained form of the given problem to find the sum of two logarithmic functions.
Then, we will get $2\left( 2\log 5+\log 4 \right)=2\log \left( 25\times 4 \right).$
We know that $25\times 4=100.$
So, we will get $2\log \left( 25\times 4 \right)=2\log 100.$
We know that $100={{10}^{2}}$ and we also have a logarithmic identity $\log {{10}^{n}}=n.$
Now, we will get $\log 100=\log {{10}^{2}}.$
Therefore, we will get $\log 100=\log {{10}^{2}}=2.$
Thus, we will get $2\log 100=2\times 2=4.$
Hence $4\log 5+2\log 4=4.$

Note: We can do this problem using an alternative method. When we use the identity $n\log x=\log {{x}^{n}},$ we will get $4\log 5+2\log 4=\log {{5}^{4}}+\log {{4}^{2}}.$ From this, we will get $4\log 5+2\log 4=\log 625+\log 16.$ When we use the identity, $4\log 5+2\log 4=\log \left( 625\times 16 \right).$ Then, we will get $\log 10000=\log {{10}^{4}}=4.$