Question

How do you calculate the expression $\log {{10}^{3}}+\log 10\left( {{y}^{2}} \right)$ ?

Answer 1

Hint: We have been given an expression which consists of two logarithmic functions out of which one term has only a constant value in the logarithm function whereas the other term has a product of a constant term and variable-y. We shall use various properties of the functions in logarithm to simplify and calculate the given expression. However, the term with only constant will be simplified completely only and the other term will be expressed in terms of variable y.

Complete step by step solution:
Given that $\log {{10}^{3}}+\log 10\left( {{y}^{2}} \right)$.
In order to simplify this expression, we shall use the basic properties of logarithmic functions.
According to a property of logarithm functions, when a logarithm function is expressed in terms of the product of n terms, then that logarithm function can be expressed as the sum of n individual logarithmic functions of those n terms, that is, $\log ab=\log a+\log b$.
Using this property on the second term of the expression, we get
$\Rightarrow \log {{10}^{3}}+\log 10\left( {{y}^{2}} \right)=\log {{10}^{3}}+\log 10+\log {{y}^{2}}$
Another property of logarithmic functions says that when an exponential term is given in the logarithmic function, then the power can be taken outside the function and entire function is modified as the product of the power of the term and the logarithm function of base, that is, $\log {{a}^{b}}=b\log a$.
Applying this property, we get
$\Rightarrow \log {{10}^{3}}+\log 10\left( {{y}^{2}} \right)=3\log 10+\log 10+2\log y$
We know that by default the logarithm function has base 10. Thus, ${{\log }_{10}}10=1$.
$\Rightarrow \log {{10}^{3}}+\log 10\left( {{y}^{2}} \right)=3\left( 1 \right)+1+2\log y$
$\Rightarrow \log {{10}^{3}}+\log 10\left( {{y}^{2}} \right)=4+2\log y$
Taking 2 common, we get
$\Rightarrow \log {{10}^{3}}+\log 10\left( {{y}^{2}} \right)=2\left( 2+\log y \right)$
Therefore, $\log {{10}^{3}}+\log 10\left( {{y}^{2}} \right)$ is calculated as $2\left( 2+\log y \right)$.

Note: We must have prior knowledge of the basic properties of the logarithmic functions. Another important property of the logarithmic functions is that when a logarithm function is expressed in terms of the division of two terms, then that logarithm function can be expressed as the difference of two individual logarithmic functions of those two terms, that is, $\log \dfrac{a}{b}=\log a-\log b$.