Question

How do you evaluate \[\log 20 + \log 50\] ?

Answer 1

Hint: In this question, we have a logarithm function. To solve this logarithm function, we used the formula. The formula is given as below. If one function is \[{\log _b}m\] and another function is \[{\log _b}n\], which is added together then it is written as below.
\[{\log _b}m + {\log _b}n = {\log _b}\left( {mn} \right)\]
Here, \[b\] is the base.

Complete step by step answer:
In this question, we used the word logarithm function. First we know about logarithm function. The logarithm function is defined as the inverse function to exponentiation. That means that in the logarithm function the number \[x\] is given as the exponent of another fixed number and the base must be raised to produce that number \[x\].
Let’s, the exponential function is denoted as below.
\[ \Rightarrow y = {e^x}\]
It is also expressed as below type.
\[y = \exp \left( x \right)\]
The above function is also called a natural exponential function.
We know that \[y = {e^x}\] is a one to one function and we also know that its inverse will also be a function. Then we find the inverse function of \[y = {e^x}\] function. The inverse function of \[y = {e^x}\]is given as below.
\[ \Rightarrow y = {\log _e}x\]
It is also written as the below type.
\[ \Rightarrow y = \ln x\]
These functions \[y = \log x\]or \[y = \ln x\] are called the logarithm function.
Now come to the question, in the question the logarithm function is given as below.
\[ \Rightarrow \log 20 + \log 50\]
We know that, if one function is \[{\log _b}m\] and another function is \[{\log _b}n\], which is added together then it is written as below.
\[ \Rightarrow {\log _b}m + {\log _b}n = {\log _b}\left( {mn} \right)\]
Here, \[b\] is the base.
In the question, the first function is \[\log 20\]and second function is \[\log 50\]. Which are added together.
Then,
\[ \Rightarrow \log 20 + \log 50\]
According to logarithm formula, It is written as,
\[\log \left( {20 \times 50} \right)\]
Then, after calculating, we get
\[ \Rightarrow \log \left( {1000} \right)\]
Then,
\[ \Rightarrow \log \left( {{{10}^3}} \right) = 3 \times \log \left( {10} \right)\]
We know that the value of \[\log \left( {10} \right)\] is \[1\].
Put this value in above equation and we get,
\[ \Rightarrow \log \left( {{{10}^3}} \right) = 3 \times \log \left( {10} \right) = 3 \times 1 = 3\]

Therefore, the result of \[\log 20 + \log 50\] is \[3\].

Note:
If you have one function is \[{\log _b}m\] and another function is \[{\log _b}n\] and you want to subtract these two logarithm functions. Then there is a formula for adding to logarithm function. The formula is given below.
\[ \Rightarrow {\log _b}m - {\log _b}n = {\log _b}\left( {\dfrac{m}{n}} \right)\]