Question

How do you write $\ln y=x\ln 2$ in exponential notation?

Answer 1

Hint: Here in this question we have been asked to write the exponential notation for given the logarithmic expression $\ln y=x\ln 2$ . From the basic definition of logarithm we know that for any exponential function given as ${{a}^{x}}=y$ the logarithmic function is expressed as ${{\log }_{a}}y=x$ . We will use this definition in order to answer this question.

Complete step by step answer:
Now considering from the question we have been asked to write the exponential notation for the given logarithmic expression $\ln y=x\ln 2$ .
From the basic definition of the logarithm which we have learnt during the basics of logarithms we know that for any exponential function given as ${{a}^{x}}=y$ the logarithmic function is expressed as ${{\log }_{a}}y=x$ .
Here as it is given as $\ln $ that means it is a natural logarithm that indicates that it has base $e$ .
Hence we can write the given expression $\ln y=x\ln 2$ in exponential notation as ${{e}^{x\ln 2}}=y$ .
We can expand this and write it as ${{e}^{x}}\left( {{e}^{\ln 2}} \right)=y$ .
Now let us evaluate the value of ${{e}^{\ln 2}}$ individually by assuming that $\ln 2=b$ we will have ${{e}^{b}}=2$ . Hence we can say that ${{e}^{\ln 2}}=2$ .
Therefore we will have $\Rightarrow 2{{e}^{x}}=y$

Therefore the exponential notation for the given logarithmic expression $\ln y=x\ln 2$ will be given as $2{{e}^{x}}=y$ .

Note: While answering questions of this type we should be sure with the logarithmic concepts that we are going to apply in between the process. This is a very simple and easy question and can be answered accurately in a short span of time. Very few mistakes are possible in questions of this type. If we had thought that this is the final exponential notation ${{e}^{x\ln 2}}=y$ which we require and not simplified it further then we may get confused if we have options and lose marks in competitive exams.