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Question

Question: How do you solve \( 7 - 2{e^x} = 5 \) ?...

How do you solve 72ex=57 - 2{e^x} = 5 ?

Explanation

Solution

Hint : In order to determine the solution of the given equation, simply it first by combining the like terms. Now We will convert the expression into logarithmic form, and to do so use the definition of exponents that the exponent of the form by=x{b^y} = x is when converted into logarithmic form is equivalent to logbX=y{\log _b}X = y ,so compare with the exponential value with by=x{b^y} = x form . And write it into logarithmic form. Remember natural logarithm of 1 is zero.

Complete step-by-step answer :
We are given an exponential equation in variable xx
72ex=57 - 2{e^x} = 5
Now combining like terms on both of the sides. Terms having tt will on the right-Hand side of the equation and constant terms on the left-hand side.
75=2ex 2=2ex   7 - 5 = 2{e^x} \\\ 2 = 2{e^x} \;
Dividing both side of the equation with number 2, we get
22=2ex2 1=ex ex=1   \dfrac{2}{2} = \dfrac{{2{e^x}}}{2} \\\ 1 = {e^x} \\\ {e^x} = 1 \;
Now we are going to convert the above exponential form into logarithmic form.
Any exponential form by=X{b^y} = X when converted into equivalent logarithmic form results in logbX=y{\log _b}X = y
So in our case we have ex=1{e^x} = 1 ,comparing it with by=X{b^y} = X , we get values of variables as
X=1 b=e y=x   X = 1 \\\ b = e \\\ y = x \;
So we have logarithmic form of our equation as
loge(1)=x\Rightarrow {\log _e}\left( 1 \right) = x
Since logarithm with base as ee is simply written as ln\ln and known as” natural logarithm”.
ln(1)=x\Rightarrow \ln \left( 1 \right) = x
Since natural logarithm of constant 1 is equal to zero, so
x=0\Rightarrow x = 0
Therefore, the solution of the given equation is x=0x = 0 .
So, the correct answer is “x=0”.

Note : 1. Don’t forgot to cross check your result
2. ln\ln is known as natural logarithm – Logarithm having base as ee
3.Logarithm of constant 1 is equal to zero.
4.Exponent and logarithm are basically inverse of each other.
5. The value of exponential constant ee is 2.718282.71828 .
6. Like terms are the terms having the same variable and power. Coefficient of the terms might be different.