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.