Question
Question: How do you solve \(3{e^x} - 2 = 0\)?...
How do you solve 3ex−2=0?
Solution
In this question, we are given an equation in terms of e (The natural base of logarithm). We have been asked to find the value of x. Start by shifting all the constants to the right-hand side of the equation. In order to simplify e, you will have to take log on both sides. After that, use certain logarithmic properties to simplify the left-hand side of the equation. Hence, you will get your answer.
Formula used:
logmn=nlogm
Complete step by step answer:
We are being given an equation in terms of natural base of logarithm, i.e., e. Let us see how we can find the value of x .
3ex−2=0 …. (given)
Step 1: In this step, we will send all the constants to the other side such that only e remains on the left-hand side.
3ex=2
Shifting the variable of ex ,
ex=32
Step 2: Now, we will take logs on both the sides of the equation.
logex=log32
Step 3: Under this step, we will apply the following property of logarithm:
logmn=nlogm
Applying on the left-hand side,
x.loge=log32
Naturally, the base of log is e. Since no base has been taken here, we will assume it to be e only.
Now, we know thatlogaa=1 . Using this in the above equation, we will put logee=1.
x(1)=log32
On simplification we get,
⇒x=log32
Hence, the value of x is log32.
Note: 1) Let us study a little more about this e.
The number of ′e′, is also known as Euler’s number. It is a mathematical constant and it has an approximate value of 2.71828. It is also the base of natural logarithm. Also, ′e′ is an irrational number, i.e., it cannot be represented in the form of qp .
2) I wrote above the identity logaa=1. Let us see from where we derived this.
We have a base conversion formula –
lognm=loganlogam
Using this formula, we can derive the formula that we used.
logaa=logaloga
Now, we will cancel out these terms and we will get our answer as 1.
Hence, logaa=1.