Question

How do you solve $3{e^x} - 2 = 0$?

Answer 1

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:
$\log {m^n} = n\log m$

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 $x$ .
$3{e^x} - 2 = 0$ …. (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.
$3{e^x} = 2$
Shifting the variable of ${e^x}$ ,
${e^x} = \dfrac{2}{3}$
Step 2: Now, we will take logs on both the sides of the equation.
$\log {e^x} = \log \left| {\dfrac{2}{3}} \right|$
Step 3: Under this step, we will apply the following property of logarithm:
$\log {m^n} = n\log m$
Applying on the left-hand side,
$x.\log e = \log \left| {\dfrac{2}{3}} \right|$
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 that${\log _a}a = 1$ . Using this in the above equation, we will put ${\log _e}e = 1$.
$x\left( 1 \right) = \log \left| {\dfrac{2}{3}} \right|$
On simplification we get,
$\Rightarrow x = \log \left| {\dfrac{2}{3}} \right|$
Hence, the value of $x$ is $\log \left| {\dfrac{2}{3}} \right|$.

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 $\dfrac{p}{q}$ .
2) I wrote above the identity ${\log _a}a = 1$. Let us see from where we derived this.
We have a base conversion formula –
$\ {\log _n}m = \dfrac{{{{\log }_a}m}}{{{{\log }_a}n}}$
Using this formula, we can derive the formula that we used.
$\ {\log _a}a = \dfrac{{\log a}}{{\log a}}$
Now, we will cancel out these terms and we will get our answer as $1$.
Hence, ${\log _a}a = 1$.