Question

Express the logarithms of the following as the sum of the logarithm $ 35 \times 46 $ .

Answer 1

Hint: Use the formula for the product of a logarithm. According to the statement, the logarithm of the product of two numbers is equal to the sum of the individual logarithms of the numbers.

Complete step-by-step answer:
Let us assume that $ a = 35 $ and $ b = 46 $
We know that as per the formula of logarithms, $ \log {\rm{ }}\left( {mn} \right) = {\rm{log m + log n}} $
We have to find the value of $ {\rm{log}}\left( {a \times b} \right) = {\rm{log}}\left( {35 \times 46} \right) $
Simplify the equation by using the formula $ \log {\rm{ }}\left( {mn} \right) = {\rm{log m + log n}} $
 $
\Rightarrow {\rm{log}}\left( {a \times b} \right) = {\rm{log}}\left( {35 \times 46} \right)\\
\Rightarrow {\rm{log}}\left( {35 \times 46} \right) = {\rm{log }}\left( {35} \right) + {\rm{log }}\left( {46} \right)
 $
We know that
 $ 35 = 5 \times 7 $ and $ 46 = 23 \times 2 $
Substitute $ 35 = 5 \times 7 $ and $ 46 = 23 \times 2 $ in the equation $ {\rm{log}}\left( {35 \times 46} \right) = {\rm{log }}\left( {35} \right) + {\rm{log }}\left( {46} \right) $
 $ \Rightarrow {\rm{log}}\left( {35 \times 46} \right) = {\rm{log }}\left( {5 \times 7} \right) + {\rm{log }}\left( {23 \times 2} \right) $
Again, use the formula $ \log {\rm{ }}\left( {mn} \right) = {\rm{log m + log n}} $ in the equation.
 $ \Rightarrow {\rm{log}}\left( {35 \times 46} \right) = {\rm{log }}\left( 5 \right) + {\rm{log }}\left( 7 \right){\rm{ + log }}\left( {23} \right){\rm{ + log}}\left( 2 \right) $
Therefore, the value of $ {\rm{log}}\left( {35 \times 46} \right) = {\rm{log }}\left( 5 \right) + {\rm{log }}\left( 7 \right){\rm{ + log }}\left( {23} \right){\rm{ + log}}\left( 2 \right) $

Additional Information:
Logarithm is a mathematical operation. It helps to find out how many times a number i.e. base is multiplied to itself in order to reach a specific number.
The logarithms whose base is $ e $ are written in the form of natural log i.e. $ \ln {\rm{ x}} $ .
When logarithms have base 10, then the base is not shown. Symbolically, the logarithm of $ a $ to the base 10 is given by $ \log {\rm{ a}} $ .
Logarithm is always present in the mathematical form of $ {\rm{lo}}{{\rm{g}}_a}b $ where $ b $ is the base and $ a $ is the argument.
Also, there is a relationship between exponential form and logarithmic form.
 $ {2^5} = 32 $ is the exponential form, then $ {\log _2}32 = 5 $ .

So, the correct answer is “Option C”.

Note: Please note that logarithm of the sum of two numbers is not equal to the sum of the logarithm of the same numbers i.e. $ {\rm{log }}\left( {a + b} \right) \ne \log {\rm{ a + log b}} $ .
Also, logarithm of the sum of the two numbers is not equal to the product of the individual logarithm of the same numbers i.e. $ {\rm{log }}\left( {a + b} \right) \ne \log {\rm{ a}} \times {\rm{log b}} $ . The logarithm of the product of the two numbers is not equal to the product of the logarithms of two numbers i.e. $ \log {\rm{ }}\left( {a \cdot b} \right) \ne \log {\rm{ a }} \times {\rm{ }}\log {\rm{ b}} $ .
These are some of the common mistakes that students make.