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Question

Question: How do you solve \(2={{e}^{5x}}\)?...

How do you solve 2=e5x2={{e}^{5x}}?

Explanation

Solution

First write the given equation in the order e5x=2{{e}^{5x}}=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\ln 2 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=e5x2={{e}^{5x}}
It can be written as e5x=2{{e}^{5x}}=2
Taking natural log both the sides, we get
lne5x=ln2\Rightarrow \ln {{e}^{5x}}=\ln 2
As, we know from the logarithmic formula lnea=a\ln {{e}^{a}}=a
So, our equation can be further simplified as
5x=ln2\Rightarrow 5x=\ln 2
Dividing both the sides by ‘5’, we get
5x5=ln25\Rightarrow \dfrac{5x}{5}=\dfrac{\ln 2}{5}
Cancelling out ‘5’ both from the numerator and the denominator, we get
x=ln25\Rightarrow x=\dfrac{\ln 2}{5}
Again as we know, the value of ln2=0.693\ln 2=0.693
So putting the value of ln2\ln 2 in the above equation, we get
x=0.6935 x=0.13863 \begin{aligned} & \Rightarrow x=\dfrac{0.693}{5} \\\ & \Rightarrow x=0.13863 \\\ \end{aligned}
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=ln25x=\dfrac{\ln 2}{5} could be the solution of the given equation, but the value of ‘x’ that we got by putting the value of ln2\ln 2 i.e. x=0.13863x=0.13863 is more appropriate.
Some basic logarithmic values should be remembered for faster and accurate calculations:
ln1=0 ln2=0.693 ln3=1.098 ln4=1.386 ln5=1.609 \begin{aligned} & \ln 1=0 \\\ & \ln 2=0.693 \\\ & \ln 3=1.098 \\\ & \ln 4=1.386 \\\ & \ln 5=1.609 \\\ \end{aligned}