Question

How do you solve $1000{{e}^{-4x}}=75$?

Answer 1

Hint: We will divide both sides of the equation by 1000. Then we will simplify the fraction on the right hand side of the equation. After that we will take the natural logarithm of the terms on both sides of the equation. Then we will divide both sides of the equation by the coefficient of $x$. This will give us the solution for the given equation.

Complete step by step answer:
The given equation is $1000{{e}^{-4x}}=75$. Let us divide both sides of the equation by 1000. We get the following,
${{e}^{-4x}}=\dfrac{75}{1000}$
Let us simplify the fraction on the right hand side of the equation. We can divide the numerator and the denominator of this fraction by 25. We know that $75\div 25=3$ and $1000\div 25=40$. Therefore, the reduced form of the fraction on the right hand side is $\dfrac{3}{40}$. So, the above equation becomes the following,
${{e}^{-4x}}=\dfrac{3}{40}$
We have the Euler's number, $e$, on the left hand side of the above equation. We know that this number is the base of the natural logarithm. Let us take the natural logarithm on both sides of the above equation. We get the following equation,
$-4x=\ln \left( \dfrac{3}{40} \right)$
We have to find the value of $x$. In the above equation, the coefficient of $x$ is $-4$. Now, we will divide both sides of the above equation by $-4$. So, we have the following,
$x=-\dfrac{1}{4}\ln \left( \dfrac{3}{40} \right)$
This is the solution of the given equation.

Note: We can use the rules of logarithmic functions to further simplify the expression. We can obtain the exact value for the variable $x$ using a calculator. After computing this value on a calculator, we obtain $x=0.6476$ as the solution. It is essential that we are familiar with the logarithm function and all the rules of this function which are useful in simplifying expressions involving the logarithm function.