Question

How do you write \[X = \log {3^8}\] in exponential form?

Answer 1

Hint: Here we will find the exponential form of the logarithm given in the question. Firstly, we will use the logarithm power rule to simplify the given logarithm. Then taking the exponential on both sides and using the exponential rule we will simplify it to get the required answer.

Complete step by step solution:
We have to change the logarithm term,
\[X = \log {3^8}\]…..\[\left( 1 \right)\]
Now using the logarithm power rule which states that \[{\log _b}\left( {{x^y}} \right) = y{\log _b}\left( x \right)\].
We will use the above rule in equation \[\left( 1 \right)\] and get,
\[ \Rightarrow X = 8\log \left( 3 \right)\]
As the above value has base as 10 , so we can write it as,
\[ \Rightarrow X = 8\ln \left( 3 \right)\]
Dividing both sides by 8, we get
\[ \Rightarrow \dfrac{X}{8} = \ln \left( 3 \right)\]
Now, taking the exponential both side, we will get
\[ \Rightarrow {e^{\dfrac{x}{8}}} = {e^{\ln \left( 3 \right)}}\]
We know that \[{e^{\ln \left( x \right)}} = x\].
Using above formula in the above equation, we get
\[ \Rightarrow {e^{\dfrac{x}{8}}} = 3\]

So, the exponential form of \[X = \log {3^8}\] is \[{e^{\dfrac{x}{8}}} = 3\] which is our desired answer.

Additional information:
Before the arrival of the calculator logarithm was a great tool to solve the bigger problems by changing the multiplication into addition which is way easier to calculate. The natural logarithm is more commonly written as \[\ln \left( x \right)\] and the natural exponential function is written as \[{e^x}\].

Note:
Logarithm states how many times we have multiplied one number to get another number. Exponent of a number means how many times we have to multiply the given number. Exponent and Logarithm undo each other and that is the fact that they work very well together. One thing that is very powerful in logarithm is that we can turn multiplication into addition and division into subtraction. One of the basic properties of numbers is that they can be written in exponential form.