Question

How do you solve $125 = {10^x}$?

Answer 1

Hint: Here in this question, we have to solve $125 = {10^x}$, the equation in the question is in the form of exponential form. We convert this exponential form equation to the logarithmic function and apply some properties of the logarithmic function and hence we determine the required solution for the question.

Complete step by step solution:
Here in this question, we have to solve $125 = {10^x}$. The RHS of the equation is in the form of exponential. The LHS of the equation is a numeral and it can be written in terms of exponential form. Let we apply the division method for the number 125, we have

5	125
5	25
5	5
	1

Therefore, the number is written as $125 = 5 \times 5 \times 5$. The exponential form of the number 125 is ${5^3}$.
Now the given equation can be written as
$ \Rightarrow {5^3} = {10^x}$
Now apply the logarithm base 10 to both sides, so we have
$ \Rightarrow {\log _{10}}{5^3} = {\log _{10}}{10^x}$
Since the above equation is in the form of $\log {m^n}$, we have property on it. It is defined as $\log {m^n} = n\log m$. By using this property, the equation is written as
$ \Rightarrow 3{\log _{10}}5 = x{\log _{10}}10$
Here we have to find the value of x , on further simplification we have
$ \Rightarrow x = \dfrac{{3{{\log }_{10}}5}}{{{{\log }_{10}}10}}$
The value of log 10 base 10 is 1. i.e., ${\log _{10}}10 = 1$
So we have
$ \Rightarrow x = 3{\log _{10}}5$
Now we determine the value of ${\log _{10}}5$ and we multiply by 3 to the result, we have
Therefore $x = 2.097$
Hence we have solved the given equation and we have found the value of x.

Note: The exponential number is defined as the number of times the number is multiplied by itself. The number which is going to multiply we call it as base and the number of times we multiply it is called as exponent. The logarithmic functions are expressed by the log function itself.