How do you write the logarithmic equation \[{{\log }_{8}}4=\dfrac{2}{3}\] in exponential form?

Question

How do you write the logarithmic equation $${{\log }_{8}}4=\dfrac{2}{3}$$ in exponential form?

Vedantu Content Team · Accepted Answer

Hint: In the above given question we can clearly see that we will have to use the basic logarithmic property which will help us convert any logarithmic function into its exponential form. The property that we are going to use to solve this question is the form in which the value on the other side of the log function will become the power to the base. Complete step-by-step solution:In the above stated question we are given with the logarithmic function which needs to be changed to its exponential form, for doing this we will use the basic form of logarithmic property which using the base to change into its logarithmic function, for this we will first understand basic things about a logarithmic function. Let us take a logarithmic function as: $${{\log }_{a}}b=c$$In this above logarithmic function “a” is the base of the function “b” is the logarithmic function value which needs to be solved with the help of logarithm and “c” is the value of the logarithm when it is solved, logarithm can also be stated as a short form of an exponential form i.e. when the base of the logarithm is shifted towards the other side of the equation i.e. when “a” is shifted towards “c” the value of logarithm function i.e. “b” becomes a to the exponential power of c i.e. $$b={{a}^{c}}$$ which is none other than exponential form. So by using this property we can use this to convert any logarithmic function into its exponential form. So now that we have learned the basics of logarithmic function we can use this to convert the given logarithmic function in the question into its exponential form.So we get the exponential form of the whole logarithmic equation to be $$4={{8}^{\dfrac{2}{3}}}$$.Note: In the above stated question we can see that how important is the basics of logarithmic function are and how it can come in handy, so learning these basic logarithmic function which are logarithm of product, logarithm of quotient and logarithm of power will only help in the long run.