Question

How do you write each number as a power of given base: 81; base 9?

Answer 1

Hint: Let’s take an example $ {{3}^{x}}=y $ then x will be equal to $ {{\log }_{3}}y $ . We can use the logarithm to solve this question we can write any positive number as a power of any positive number.

Complete step by step answer:
Let’s try to understand the concept of the logarithm
If $ {{a}^{x}}=b $ then $ x={{\log }_{a}}b $
a and b should be positive numbers. if a or b will be negative numbers then the graph of logarithm will not be continuous.
If we try to write 9 as the power of base 3 it will be 2 which is $ {{\log }_{3}}9=2 $
If we write b as a power of a then it will be $ {{\log }_{a}}b $ , $ {{a}^{{{\log }_{a}}b}}=b $
So if we write 81 as a power given base 9 the answer will be $ {{\log }_{9}}81 $ which is equal to 2.
If we write any positive integer x as the power of base 9 the answer is $ {{\log }_{9}}x $
 $ {{\log }_{a}}b $ will be a positive if both a and b will be greater than 1 or both a and b will be less than 1.
 $ {{\log }_{a}}b $ is a negative when one number is greater than 1 and one is less than 1.
For example $ {{\log }_{0.5}}0.3 $ and $ {{\log }_{3}}5 $ will be positive.
 $ {{\log }_{0.4}}5 $ and $ {{\log }_{3}}0.2 $ will be negative.

Note:
The domain of the function $ y=\log x $ is always a positive number. The function will not exist if the domain will be negative because the power of any positive integer will always positive and the base can’t be negative because the graph will not be continuous. It will be discrete.