Question

How do you solve $\dfrac{500}{100-{{e}^{\dfrac{x}{2}}}}=20$ ?

Answer 1

In this question, we have to simplify the equation and find the value of x. There is use of natural logarithm ‘e’ in the given question. We have to use some important properties of natural logarithm. They are:
Product rule: ln(xy) = ln(x) + ln(y)
Quotient rule: $\ln \left( \dfrac{x}{y} \right)$ = ln(x) – ln(y)
Power rule: $\ln \left( {{x}^{n}} \right)=n\ln \left( x \right)$

Complete step-by-step solution:
Now, let’s discuss the question.
As we know that logarithm is defined as the power to which number must be raised to get some other values and it is the most convenient way to express large numbers. There are basically two different types of logarithms. First is the common logarithm and the other one is natural logarithm. The common logarithm is known as the base 10 logarithm. It is represented as log10 or simply log whereas natural logarithm is called the base e logarithm. The natural logarithm is represented as ln or log(e). ‘e’ is the Euler’s constant which is equal to 2.71828. the value of ln(e) is 1.
Let’s study some basic rules for natural logarithm.
Product rule: ln(xy) = ln(x) + ln(y)
Quotient rule: $\ln \left( \dfrac{x}{y} \right)$ = ln(x) – ln(y)
Power rule: $\ln \left( {{x}^{n}} \right)=n\ln \left( x \right)$
Write the equation given in question.
$\Rightarrow \dfrac{500}{100-{{e}^{\dfrac{x}{2}}}}=20$
Move the terms of denominator to the other side in product with 20:
$\Rightarrow 500=20\times \left( 100-{{e}^{\dfrac{x}{2}}} \right)$
Open brackets:
$\Rightarrow 500=20\times 100-20\times {{e}^{\dfrac{x}{2}}}$
Multiply the terms:
$\Rightarrow 500=2000-20{{e}^{\dfrac{x}{2}}}$
Take constant terms on one side of the equation:
$\begin{aligned} & \Rightarrow 20{{e}^{\dfrac{x}{2}}}=2000-500 \\\ & \Rightarrow 20{{e}^{\dfrac{x}{2}}}=1500 \\\ \end{aligned}$
Take 20 to the other side:
$\Rightarrow {{e}^{\dfrac{x}{2}}}=\dfrac{1500}{20}$
Reduce the fraction:
$\Rightarrow {{e}^{\dfrac{x}{2}}}=75$
Take natural log on both the sides:
$\Rightarrow \ln \left( {{e}^{\dfrac{x}{2}}} \right)=\ln \left( 75 \right)$
The value for ln(75) is 4.32.
$\Rightarrow \ln \left( {{e}^{\dfrac{x}{2}}} \right)=$ 4.32
Now, apply the property of natural log:
Power rule: $\ln \left( {{x}^{n}} \right)=n\ln \left( x \right)$
$\Rightarrow \dfrac{x}{2}=4.32$
Solve for x:
$\therefore$x = 8.64
This is the final answer.

Note: All you need to do is check each term of the expression if any property is applicable there or not. Then only apply the property on the whole expression, otherwise first you need to solve each term to its end and then apply the property on the whole expression.