Question

How do you solve \[lo{{g}_{x}}1000=3\]?

Answer 1

Hint: Exponential function is one-to-one function. So, its inverse exists and its inverse is called logarithm. Let the exponential function be given by ${{a}^{y}}$ and let the value of ${{a}^{y}}$ be $x$. That is ${{a}^{y}}=x$. Then we can find the inverse of this which can be given by
$y={{a}^{x}}$. ‘a’ is the base, $x$ is the power and $y$ is the number. Then we can find the logarithmic function \[x=lo{{g}_{a}}\text{ }y\]for the function ${{a}^{x}}=y$. Its base is $a$. There are certain conditions on the base that is the base should be greater than $0$ and it should not be equal to $1$.

Complete step by step solution:
We have to solve \[lo{{g}_{x}}1000=3\]
That is we have to find the value of $x$
We can write down this logarithmic function in the exponential form as
\[lo{{g}_{x}}1000=3\]
=> ${{x}^{3}}=1000$
Now since we can write $1000$ as ${{10}^{3}}$
Therefore ${{x}^{3}}={{10}^{3}}$
We know that if the exponential value is equal and the exponents are also equal then the base should also be equal.
So, we get \[x=10\]

Thus \[x=10\] is the solution of the equation \[lo{{g}_{x}}1000=3\].

Note:
A logarithmic function with base e is called the natural logarithmic function. The log of $1$ to any base gives $0$.There are many properties of logarithm which are very useful in solving the problems. We generally use the logarithmic function with base $10$ as it is the most common logarithmic function.