Question

Find the value of x if \[{{\log }_{x}}625={{\log }_{10}}100\] .

Answer 1

Hint: Now we know that ${{\log }_{a}}b=n$ means nothing but ${{a}^{n}}=b$ . Hence using this on both sides we will form an equation of exponent. Now taking the square root on both sides we will find the value of x.

Complete step-by-step answer:
Now before solving the equation let us understand Logarithms.
Logarithms are nothing but a way for writing exponents.
We know what exponents are.
$10\times 10\times 10\times 10$ can be written as ${{10}^{4}}$ .
Similarly we can we write $2\times 2\times 2={{2}^{3}}$
An exponent is nothing but a number with power. Even 2 can be written in exponent form just the power will be 1. Hence ${{2}^{1}}=2$ .
Now let us consider any exponent equation say ${{a}^{n}}=b$ .
Then we say that ${{\log }_{a}}b=n$ where a is called the base of log, b is called the argument and n is called the exponent of log.
Now in logarithm the base of log and the argument is always positive and the base is never equal to 1.
For example consider ${{10}^{2}}=100$ .
This can be written in logarithmic form as ${{\log }_{10}}100=2$
Let us consider the given equation \[{{\log }_{x}}625={{\log }_{10}}100\]
Now if we have ${{a}^{n}}=b$ , then we say that ${{\log }_{a}}b=n$
Hence we get that ${{\log }_{10}}100$ is equal to 2, since ${{10}^{2}}=100$ .
Now let us substitute ${{\log }_{10}}100=2$ in the equation. Hence we get, ${{\log }_{x}}625=2$ .
Now again we have ${{a}^{n}}=b$ . Then ${{\log }_{a}}b=n$
Hence ${{\log }_{x}}625=2$ can be written as ${{x}^{2}}=625$ .
Now taking the square root on both sides we get $x=\sqrt{625}$ .
Hence x = 25 or x = -25.
But since x is the base of logarithm hence it cannot be negative. Hence the only value of x is 25.

Note: Note that we have property of logarithm which says ${{\log }_{a}}{{b}^{b}}=n{{\log }_{a}}b$ hence we get ${{\log }_{10}}100={{\log }_{10}}{{10}^{2}}=2{{\log }_{10}}10$ . Now again we know that ${{\log }_{a}}a=1$ this means ${{\log }_{10}}10=1$ . Hence we have ${{\log }_{10}}100=2$ .