Question

How do you write ${{\log }_{5}}\left( 625 \right)=x$ in exponential form.

Answer 1

Hint: Now we know that according to the definition of logarithm function if we have an equation of the form ${{\log }_{a}}b=x$ then we can write it as ${{a}^{x}}=b$ . Hence we will compare the values and find a, b and x and then write the equation in the required form using definition.

Complete step by step solution:
Now let us first understand the function Log. A function in log is written in the form ${{\log }_{a}}\left( b \right)$ . Here a is a fixed number and called the base of logarithm. Now b is the variable in the function. Now note that b > 0 also we have $a>0$ and $a\ne 1$. Now let us understand the meaning of ${{\log }_{a}}\left( b \right)$ the value gives the exponent to which a must be raised to get b. Now let us say the value of ${{\log }_{a}}\left( b \right)=x$ then we have ${{a}^{x}}=b$ .
Now let us learn a few basic properties of log.
The power rule${{\log }_{a}}{{b}^{n}}=n{{\log }_{a}}b$
Rule for multiplication $\log \left( ab \right)=\log \left( a \right)+\log \left( b \right)$
Rule for division$\log \left( \dfrac{a}{b} \right)=\log \left( a \right)-\log \left( b \right)$
Now consider the given equation. We have ${{\log }_{5}}625=x$ .
Now we know by definition of log that the equation means ${{5}^{x}}=625$ .
Hence the equation in exponent form can be written as ${{5}^{x}}=625$ .
Now we can also find the solution of the equation as we know that ${{5}^{4}}=625$ .

Note: Now note that many times the base of log is not written in such cases the base of log is assumed to be 10. This is the general base of log. Also sometimes we write log as ln. This is called the natural logarithm and it has base as e = 2.7182….