Question

How do you find $x$ for ${\log _5}x = 4?$

Answer 1

Hint: This problem deals with logarithms. This problem is rather very easy and very simple, though it seems to be complex. In mathematics logarithms are an inverse function of exponentiation. Which means that the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x.
If given ${\log _b}x = a$, then $x$ is given by :
$ \Rightarrow x = {b^a}$

Complete step-by-step solution:
Here given the equation of logarithmic expression, which is ${\log _5}x = 4$.
The above equation is a logarithm expression of $x$ to the base 5, which is equal to 4.
Considering the given equation of logarithmic expression, as given below:
$ \Rightarrow {\log _5}x = 4$
Now as already discussed above in the previous section, if any logarithmic expression of $x$, to the base of b is equated to a, then the value of $x$ is equal to the b raised to the exponent of a.
That is the expression of $x$ here is equal to b , which has the power, also called the exponent which is a.
Similarly in this given expression of logarithm also, the expression of $x$ would be equal to the number 5 raised to the power of its logarithmic base which is 4.
From this we can write the expression of $x$, as given below:
$ \Rightarrow x = {\left( 5 \right)^4}$
$ \Rightarrow x = {5^4} = 625$
Thus the value of the expression of $x$, is given below:
$\therefore x = 625$

The value of x is equal to 625.

Note: Please note that while solving any problem based on logarithms, some important formulas based on logarithms are used. The important logarithmic basic formulas are used anywhere such as:
$ \Rightarrow {\log _{10}}\left( {ab} \right) = {\log _{10}}a + {\log _{10}}b$
$ \Rightarrow {\log _{10}}\left( {\dfrac{a}{b}} \right) = {\log _{10}}a - {\log _{10}}b$
$ \Rightarrow $If ${\log _e}a = b$, then $a = {e^b}$
Hence $a = {e^{{{\log }_e}a}}$, since $b = {\log _e}a$.