Question

How do you solve $500{e^{ - {\text{x}}}} = 300$?

Answer 1

Hint: In this question, we are asked to solve $500{e^{ - {\text{x}}}} = 300$ . As this equation involves exponential, to derive the accurate answer we would require a calculator. First, we need to eliminate the numbers or coefficient of the exponential and then take the log in order to find its value.

Formula used: Properties of log and exponentials:
$\ln {\text{ }}{{\text{x}}^{\text{a}}} = {\text{ a ln x}}$
$\ln e = 1$

Complete step-by-step solution:
The given equation is $500{e^{ - {\text{x}}}} = 300$ . We need to find the value of ${\text{x}}$ .
First, we will eliminate the coefficient of the exponential,
$500{e^{ - {\text{x}}}} = 300$
By dividing $500$ on both the sides, we get,
$\Rightarrow$$\dfrac{{500{e^{ - {\text{x}}}}}}{{500}} = \dfrac{{300}}{{500}}$
On simplifying, it becomes,
$\Rightarrow$${e^{ - {\text{x}}}} = \dfrac{3}{5}$
Now, we have to take the natural log on both the side to eliminate the exponential in order to separate the variable x,
Taking natural log on both the sides we get,
$\Rightarrow$$\ln \left( {{e^{ - {\text{x}}}}} \right) = \ln \left( {\dfrac{3}{5}} \right)$
Using the properties of natural log, $\ln {\text{ }}{{\text{x}}^{\text{a}}} = {\text{ a ln x}}$ , we get,
$\Rightarrow$$( - {\text{x) }}\ln e = \ln \left( {\dfrac{3}{5}} \right)$
Again, using the property of log and exponential, ( $\ln e = 1$ )
$\Rightarrow$$ - {\text{x = ln}}\left( {\dfrac{3}{5}} \right)$
By multiplying $ - 1$ on both sides we get,
$\Rightarrow$${\text{x = - ln}}\left( {\dfrac{3}{5}} \right)$
This can be kept as it is or we can also remove the minus symbol by reciprocal of the terms.

Therefore, ${\text{x = ln}}\left( {\dfrac{5}{3}} \right)$ is the required answer.

Note: Students can find the accurate answer by using the scientific calculator, that is ${\text{x = ln 5 - ln 3 = }}1.6094 - {\text{ }}1.0986 = 0.5108$
Further note that,
The mathematical constant $e$ is an exponential function which is a unique function that equals its own derivative. It is the base of nature logarithm.
Properties of $e$ and $\ln $ are given below,
For any product of $\ln $ , the values will be added
\[\ln \left( {{\text{xy}}} \right) = \ln \left( {\text{x}} \right) + \ln \left( {\text{y}} \right)\]
For any division, the values will be subtracted
\[\ln \left( {\dfrac{{\text{x}}}{{\text{y}}}} \right) = \ln \left( {\text{x}} \right) - \ln \left( {\text{y}} \right)\]
The log of power property
\[\ln \left( {{\text{xy}}} \right) = {\text{y}}\ln \left( {\text{x}} \right)\]
Log of e
\[{\mathbf{ln}}\left( {\mathbf{e}} \right) = 1\]
Also, $\ln {\text{ }}{{\text{x}}^{\text{a}}} = {\text{ a ln x}}$
These are the important properties of log which will be frequently asked or used.