Question
Question: How do you solve \( 7 = 5{e^{0.2x}} \) ?...
How do you solve 7=5e0.2x ?
Solution
Hint : In order to determine solution of the given equation, simply it first by dividing both sides by 5 .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 is when converted into logarithmic form is equivalent to logbX=y ,so compare with the given logarithm value with this form . And write it into logarithmic form and solve the equation for x to get the required result.
Complete step-by-step answer :
We are given an exponential equation in variable x
7=5e0.2x
Simplifying the equation by dividing both sides of the equation with 5 , we get
51×7=51×5e0.2x 57=e0.2x
Now we are going to convert the above exponential form into logarithmic form.
Any exponential form by=X when converted into equivalent logarithmic form results in logbX=y
So in our case we have 57=e0.2x ,comparing it with by=X , we get values of variables as
X=57 b=e y=0.2x
So we have logarithmic form of our equation as
⇒loge57=0.2x
Since logarithm with base as e is simply written as ln and known as” natural logarithm”.
⇒ln57=0.2x
Now dividing both sides of the equation with 0.2 , we have value of x as
⇒0.21ln57=0.20.2x ⇒5ln57=x ⇒x=5ln57
Therefore, the solution of the given equation is x=5ln57 .
So, the correct answer is “x=5ln57”.
Note : 1. A logarithm is basically the reverse of a power or we can say when we calculate a logarithm of any number , we actually undo an exponentiation.
2.Any multiplication inside the logarithm can be transformed into addition of two separate logarithm values .
logb(mn)=logb(m)+logb(n)
3. Any division inside the logarithm can be transformed into subtraction of two separate logarithm values .
logb(nm)=logb(m)−logb(n)
4. Any exponent value on anything inside the logarithm can be transformed and moved out of the logarithm as a multiplier and vice versa.
nlogm=logmn