Question
Question: How do you solve \(2={{e}^{5x}}\)?...
How do you solve 2=e5x?
Solution
First write the given equation in the order e5x=2. Then take natural log on both the sides to remove the ‘e to the power’ part. Then do the necessary simplification to get the value of ‘x’ by putting the value of ln2 at last.
Complete step-by-step answer:
Solving the equation means, we have to find the value of ‘x’ for which the equation gets satisfied.
Considering our equation 2=e5x
It can be written as e5x=2
Taking natural log both the sides, we get
⇒lne5x=ln2
As, we know from the logarithmic formula lnea=a
So, our equation can be further simplified as
⇒5x=ln2
Dividing both the sides by ‘5’, we get
⇒55x=5ln2
Cancelling out ‘5’ both from the numerator and the denominator, we get
⇒x=5ln2
Again as we know, the value of ln2=0.693
So putting the value of ln2 in the above equation, we get
⇒x=50.693⇒x=0.13863
This is the required solution of the given question.
Note: Taking the natural log on both the sides should be the first approach for solving such questions. Some basic logarithmic rules and values should be known for maximum simplification. For example, x=5ln2 could be the solution of the given equation, but the value of ‘x’ that we got by putting the value of ln2 i.e. x=0.13863 is more appropriate.
Some basic logarithmic values should be remembered for faster and accurate calculations:
ln1=0ln2=0.693ln3=1.098ln4=1.386ln5=1.609