Question

How do you write \[{{\log }_{216}}36=\dfrac{2}{3}\] in exponential form?

Answer 1

Hint: The logarithmic expression \[{{\log }_{a}}b=m\], here \[a\And b\in \] positive real numbers, and a is not equals to 1. The argument of logarithm is b, the base of the logarithm is a, and m is called the value of the logarithm. We should know that the logarithmic and exponential forms have only different structures but they express the same meaning. To write this in exponential form, it can be done as \[{{a}^{m}}=b\].

Complete step by step answer:
We are given the expression \[{{\log }_{216}}36=\dfrac{2}{3}\], we can see that this is a logarithmic expression of the form \[{{\log }_{a}}b=m\]. We have to write this in exponential form. Comparing this expression with the general form of the logarithm which is, \[{{\log }_{a}}b=m\], we get the base of logarithm a= 216, the argument of logarithm b = 36, and the value of logarithm m = \[\dfrac{2}{3}\].
The logarithmic expression \[{{\log }_{a}}b=m\] is written in exponential form as, \[{{a}^{m}}=b\].
Substituting the value of the base, argument, and the value we have in the above expression. We get the exponential form as,
\[{{216}^{\dfrac{2}{3}}}=36\]

Hence, the exponential form of the logarithmic expression \[{{\log }_{216}}36=\dfrac{2}{3}\] is \[{{216}^{\dfrac{2}{3}}}=36\] .

Note: It should be noted that both the logarithmic form and exponential form express the same meaning. The exponential form states that the \[{{\dfrac{2}{3}}^{th}}\] power of 216 is 36. \[{{\dfrac{2}{3}}^{th}}\]power means square of the cube root. The logarithmic form of the expression states that 216 should be raised to power \[\dfrac{2}{3}\], for it to become equal to 36. One should know how to convert exponential to logarithmic form too, as it can also be asked.