How do you solve\[{e^{2x}} - 3{e^x} - 4 = 0?\]

Hint: This question involves the arithmetic operations like addition/ subtraction/ multiplication/ division. We need to know the relation between exponent and natural logarithm. Also, we need to know the formulae or conditions involved with natural logarithm and exponential terms. We need to know how to calculate the natural logarithm in the scientific calculator.

Complete step-by-step solution:
The given equation in the question is shown below,
\[{e^{2x}} - 3{e^x} - 4 = 0\]
The above equation can also be written as,
\[{e^{2x}} - 3{e^x} = 4\]\[ \to equation\left( 1 \right)\]
(When\[ - 4\]is the move from LHS to RHS of the equation it convert into\[4\])
For simplifying the equation\[\left( 1 \right)\], we multiply the term\[\ln \]on both sides of the equation\[\left( 1 \right)\]. So, we get
\[equation\left( 1 \right) \to {e^{2x}} - 3{e^x} = 4\]
\[\ln {e^{2x}} - 3\ln {e^x} = \ln 4 \to equation\left( 2 \right)\]
We know that,
\[\ln {x^n} = n\ln x \to equation\left( 3 \right)\]
By using the equation\[\left( 3 \right)\] in the equation\[\left( 2 \right)\] we get,
\[2x\ln e - 3x\ln e = \ln 4 \to equation\left( 4 \right)\]
We know that,
\[\ln = \dfrac{1}{e}\]
So,
\[\ln e = 1\]
By using the above condition in the equation\[\left( 4 \right)\]we get,
\[equation\left( 4 \right) \to 2x\ln e - 3x\ln e = \ln 4\]
\[
  2x\left( 1 \right) - 3x\left( 1 \right) = \ln 4 \\
  2x - 3x = \ln 4 \\
 \]
Here, \[2x - 3x = - x\]
So, we get
\[ - x = \ln 4\]
Let’s multiply\[ - \]on both sides of the above equation, we get
\[x = - \ln 4\]
So, the final answer is,
\[x = - \ln 4\]
By using the scientific calculator we get,
\[x = - \ln 4 = - 1.3863\]

Note: This question involves the arithmetic operations like addition/ subtraction/ multiplication/ division. We would remember the basic conditions related to\[\ln \]and\[e\]. To solve these types of questions we would perform arithmetic operations with terms which have different signs. So, we would remember the following things,
When a negative term is multiplied with a negative term the final answer will be a positive term.
When a positive term is multiplied with a positive term the final answer will be a positive term.
When a negative term is multiplied with a positive term the final answer will be a negative term.