Question

How do you solve ${e^x} = 4$ ?

Answer 1

Hint: The equation given in the question is an exponential equation that is defined as the equation in which one term is raised to the power of another term, for example $a = {x^y}$. So, the given equation is an exponential function. To solve such functions, we use the concept of logarithm functions, the logarithm functions are of the form $y = {\log _x}a$ . When the base of a logarithm function (x) is equal to e then the function is written as $\ln a$ , it is called the natural logarithm. $e$ is a mathematical constant and is irrational and transcendental. Its value is nearly equal to $2.718281828459$ . Using this information, we can find out the correct answer.

Complete step by step solution:
We have to solve ${e^x} = 4$
Taking the natural log on both sides –
$\ln {e^x} = \ln 4$
From the laws of the logarithm, we know that –
$
   \Rightarrow \log {x^n} = n\log x \\
   \Rightarrow \ln {e^x} = x\ln e = x\,\,\,(\ln e = 1) \\
 $
Using this value in the above equation, we get –
$
   \Rightarrow x = \ln 4 \\
   \Rightarrow x = 1.38629436 \\
 $
But we take the values up to 3 decimal places, so we round off the above answer.
$ \Rightarrow x = 1.386$
Hence when ${e^x} = 4$ , $x = 1.386$

Note: These types of questions are usually solved by factoring the non-exponential side of the equation. We write that number as a product of the prime numbers and compare the obtained base obtained with the base of the power x. For example, let ${6^x} = 216$ , we factorize 216 as – $216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3$ , now we write this number as a product of the base number – $216 = 6 \times 6 \times 6 = {6^3}$ , on comparing the left-hand side and the right-hand side of the example, we get x=3. But in the given question, when we factorize 4, we get – $4 = 2 \times 2 = {2^2}$ but $2 \ne e$ so we cannot use this method and we have to follow the above-mentioned solution for solving similar questions.