Question

How do you write \[{\log _2}\left( {64} \right) = 6\] in exponential form?

Answer 1

Hint:We need to know how to expand the \[{2^n}\] terms. Also, we need to know the basic conditions in logarithmic functions. Also, we need to compare the given equation with the logarithmic conditions. This question involves the arithmetic operations of addition/ subtraction/ multiplication/ division to solve the given question. The final answer would be a most simplified form of the given equation.

Complete step by step solution:
The given equation is shown below,
\[{\log _2}\left( {64} \right) = 6 \to \left( 1 \right)\]
We know that,
\[{\log _a}x = y \Rightarrow {a^y} = x \to \left( 2 \right)\]
To solve the given equation we would compare the given equations with the equation \[\left( 2 \right)\].
Let’s compare the equation \[\left( 1 \right)\] and \[\left( 2 \right)\], we get,
\[\left( 1 \right) \to {\log _2}\left( {64} \right) = 6\]
\[\left( 2 \right) \to {\log _a}x = y \Rightarrow {a^y} = x\]
So, we get the value of \[x,y\] and \[a\]. So, the value of\[a\] is \[2\] the value of\[x\]is\[64\] and the value of\[y\]is\[6\].
Let’s substitute these values in the equation \[\left( 2 \right)\],
we get
\[
\left( 2 \right) \to {\log _a}x = y \Rightarrow {a^y} = x \\
{\log _2}64 = 6 \Rightarrow {2^6} = 64 \\
\]
Let’s verify the above equation,
\[{2^6} = 2 \times 2 \times 2 \times 2 \times 2 \times 2\]
\[
{2^6} = 4 \times 4 \times 4 \\
{2^6} = 16 \times 4 \\

{2^6} = 64 \\
\]
Verified.
So, the final answer is, \[{2^6} = 64\]

Note: In this type of question we would involve the operation of addition/ subtraction/ multiplication/ division. Note the \[{2^n}\] means \[n\] times we have to multiply the term \[2\]. So, \[{2^6}\] we had multiplied \[6\] times the term \[2\]. We can also use a scientific calculator to find the value of \[{2^n}\] terms while solving the given question. Remember the basic logarithmic conditions to make easy calculations. The final answer would be in exponential form because it was mentioned in the given question. For solving these types of questions we would compare the given question with the relatable logarithmic conditions. The final answer would be a most simplified form of a given question.