Question

If it is given that ${{\log }_{10}}x=a$ and ${{\log }_{10}}y=b$ , then find the value of $xy$.

Answer 1

Hint:First we will define what a logarithmic function is and then we will by definition write the given log functions into the power form. Then as we require we will multiply the value of $x$ and $y$ and then get our answer. We will add the powers as their bases will be the same.

Complete step by step answer:
Let’s understand the meaning of logarithmic functions. A logarithm is defined as the power to which number must be raised to get some other values. It is the most convenient way to express large numbers. A logarithm has various important properties that prove multiplication and division of logarithms can also be written in the form of logarithm of addition and subtraction.
The logarithm of a positive real number $a$ with respect to base $b$ , a positive real number, is the exponent by which $b$ must be raised to yield $a$ that is ${{b}^{y}}=a$ and it is read as “the logarithm of $a$ to base $b$ .” It is also written as: ${{b}^{y}}=a\Leftrightarrow {{\log }_{b}}a=y$
We are given in the question that \[{{\log }_{10}}x=a\] and ${{\log }_{10}}y=b$
Now, according to the definition of logarithm:
${{\log }_{10}}x=a\Rightarrow x={{10}^{a}}\text{ }.............\text{ Equation 1}$ and ${{\log }_{10}}y=b\Rightarrow y={{10}^{b}}\text{ }.............\text{ Equation 2}$
We have to find the value of $xy$, now we will use the values from equation 1 and equation 2 :
$xy={{10}^{a}}\times {{10}^{b}}$ , Now we know that if the bases are same then their power is added $={{10}^{\left( a+b \right)}}$
Therefore, the answer is ${{10}^{\left( a+b \right)}}$

Note:
 When the base is not mentioned in the logarithmic function, we automatically take the base as $e$ and it is written as ${{\log }_{e}}m\text{ or }\ln m$ both of which means the same thing. Another definition that can be said for logarithm is that logarithms are the inverse process of the exponentiation.