Question

How do solve ${{5}^{x}}=625$?

Answer 1

Hint: The equation given in the above question is a linear equation in one variable, that is x. The question says that we have to solve the given equation in x. In other words, we have to find the values of x which will satisfy the given equation. Try to find the value of x by performing some mathematical operations.

Complete step by step solution:
The given equation says that ${{5}^{x}}=625$ … (i)
The given expression is a function of x in which a constant number is raised to the variable x. This time of function is called an exponential function.
The general form of an exponential function with variable x is given as $f(x)={{a}^{x}}$, where f(x) is a function of x and a is the constant. The variable x is called exponent and the constant ‘a‘ is referred to as the base of the exponent.
In the given equation, the base of the exponent x is equal to 5 and in the equation it is shown that 5 raised to x is equal to 625. This means that we have to find the value of x such that when this value is used as the exponent for base 5, it becomes equal to 625.
From the table of squares, we know that 625 is a square of 25 and 25 is a square of 5.
Then,
$\Rightarrow 625=5\times 5\times 5\times 5$.
$\Rightarrow {{5}^{4}}=625$ … (ii)
By comparing (i) and (ii) we get that $x=4$.

Note: We can solve the given equation with another method if we know about logarithmic function and its properties. Logarithmic function is the inverse of exponential function. For exponential function $y={{a}^{x}}$, the corresponding logarithmic function is $x={{\log }_{a}}y$.
If you know about logarithmic properties then you can solve this equation.