Question

How do you write ${10^3} = 1000$ in log form?

Answer 1

Hint: Let us first know what actually log is. The logarithm of a log is the power to which a number should be raised so as to get another number. It simply answers the question of how many of the given numbers shall we multiply to get another number. If we want to express the fact that any number $a$ shall be multiplied $y$ times in order to get some number $x$, it can be done in the following way:
${a^y} = x \Leftrightarrow y = {\log _a}x$
Here $a$ is known as the base of the logarithm and $y$ is known as the exponent. We will use this expression to solve the question given above.

Complete Step by Step Solution:
The number of which the logarithmic form has to be found out is $1000$.
We can write it in the form of ${a^y} = x$, such that
$ \Rightarrow {10^3} = 1000$ where $a = 10$; $y = 3$ and $x = 1000$.
It must be known that ${a^y} = x \Leftrightarrow y = {\log _a}x$. We are given with the form on the left hand side of the equation i.e. the exponential form and we have to find the logarithmic form, i.e. on the right hand side of the equation. Therefore on substituting the obtained values of $a$, $y$ and $x$ in the expression $y = {\log _a}x$, we will get
$ \Rightarrow 3 = {\log _{10}}1000$
On rearranging, we get
$ \Rightarrow {\log _{10}}1000 = 3$

Hence, ${10^3} = 1000$ can be written in log form as ${\log _{10}}1000 = 3$.

Note:
It must be remembered that Log and log are not the same, i.e. $\log \ne Log$. Also, we know that whatever be the exponent of 0, it will remain 0 only. But still, $\log $with base $0$ is not defined.