Question

How do you write \[{{\log }_{4}}64=3\] in exponential form?

Answer 1

Hint: As we can see that the above equation is a logarithmic equation. In order to write it in exponential form we have to get a basic definition of logarithms. First, we will try to condense the log expression into simple logarithms then we will use the rules to isolate the logarithmic expressions which have the same bases on both sides of the equation. Thus, taking reverse of it will give us the equation in exponential form.

Complete step-by-step solution:
This question belongs to the concept of solving logarithmic equations or functions. Logarithmic equations involve logarithm of an expression logarithm is just the opposite or inverse of exponentiation. Thus, we can conclude that the logarithm of a given function is the exponent to which another number must be raised in order to get the original number and in order to convert a logarithmic function into exponential form we have to take inverse.
Now in the question we have \[{{\log }_{4}}64=3\] . Here we can see that the expression is a simple logarithmic equation therefore in order to convert it into exponential form we will go with the basic definition of logarithms.
First, we will rewrite the given equation in exponential form as per the definition of logarithm states that if x and z are positive real numbers also z is not equal to zero then \[b={{\log }_{z}}x\] can be written as \[{{z}^{b}}=x\]. It is an exponential function.
Therefore,
\[\begin{align}
  & {{\log }_{4}}64=3 \\
 & \Rightarrow 64={{4}^{3}} \\
\end{align}\]
Or \[{{4}^{3}}=64\]
Now we can see that on both sides the values are same thus \[L.H.S=R.H.S\]
Therefore, the given logarithmic function can be expressed into exponential form as \[{{4}^{3}}=64\].

Note: While solving the question keep in mind that logarithmic and exponential rules, both have different and reverse of each other. We will never get a negative value after solving a logarithmic equation. Do check your answer is whether it is in the form of \[{{a}^{x}}=b\].