Question

How do you write each number as a power of the given base: -64, base -4?

Answer 1

Hint: As the base and the value is given, we will use the logarithm to solve the question. As we can write any positive number as a power of any positive number and any negative number as a power of any negative number.
Let’s take an example ${3^x} = y$, then x will be equal to ${\log _3}y$.

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 both should be either positive or negative numbers. If a or b will be negative numbers then the graph of logarithm will not be continuous.
If we try to write -4 as the power of base -4 it will be 1 which is ${\log _{ - 4}} - 4 = 1$.
If we write b as the power of a then it will be ${\log _a}b$, ${a^{{{\log }_a}b}} = b$.
So, if we write -64 as a power given base -4 the answer will be ${\log _{ - 4}} - 64$ which is equal to 3.
Then,
$ - 64 = {\left( { - 4} \right)^3}$

Hence, -64 as a power of the base -4 is 3.

Additional Information:
\[{\log _a}b\] will be 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 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 be positive and the base can’t be negative because the graph will not be continuous. It will be discrete.