Question

Solve the exponential equation algebraically. Approximate the result to three decimal places.$\frac{500}{100-e^{x/2}}=20$

Short answer

Using a calculator to numerically solve $x=2\ast \ln (75)$, the answer is $x\approx 8.317$.

Step-by-step solution

- Simplify the Equation

Start by multiplying both sides by $(100-e^{x/2})$ to cancel the denominator on the left side. This will leave us with: $$500=20\ast (100-e^{x/2})$$

- Expand and simplify

Now, expand the right side of the equation and then simplify by subtracting 500 from both sides: $$2000-20\ast e^{x/2}=500,whichleadsto-20\ast e^{x/2}=500-2000,-20\ast e^{x/2}=-1500$$

- Isolate the Exponential Term

In order to isolate $e^{x/2}$, divide both sides of equation by -20: $$e^{x/2}=75$$

- Apply Natural Logarithm on Both sides

We need to get $x$ out of the exponent. To accomplish that, apply natural logarithm $\ln$ on both sides. Remember the property of logarithm: $\ln e^m=m$: $$\ln (e^{x/2})=\ln (75),whichleadsto\frac{x}{2}=\ln (75)$$

- Solve for x

Finally, to get $x$, we multiply both sides by 2. Solving this equation will provide the numerical value and rounding it to three decimal places will give the final answer. $$x=2\ast \ln (75)$$

Key concepts

Solving Exponential Equations Algebraically

Solving exponential equations algebraically involves finding the value of the variable when it is in the exponent of a base. To handle these equations, the focus must be on isolating the exponential term. In our example, we started by eliminating the fraction and then expanded our terms to isolate the exponential part. This is a crucial step, as having the exponential term on one side helps to apply logarithmic functions to bring down the exponent and solve for the variable.

One common method for solving exponential equations algebraically is to use logarithms. This stems from their unique property, where applying a logarithm to both sides of an equation allows one to move from exponential to linear expressions. This process transforms a potentially difficult problem into one that is often much more manageable. For example, when we encounter an equation such as $e^{x/2}=75$, by applying the natural logarithm, we are able to move forward by converting the exponential equation into a linear one.

Natural Logarithm Properties

The natural logarithm, often denoted as $\ln$, is a logarithm with base $e$, where $e$ is an irrational and transcendental number approximately equal to 2.71828. Understanding the properties of natural logarithms can demystify solving exponential equations.

One of the primary properties we used in our exercise is $\ln (e^m)=m$, a direct consequence of the fact that $e$ and $\ln$ are inverse functions. The natural logarithm can also be utilized to solve equations with differing bases by using the change of base formula: $\ln (a^b)=b\ast \ln (a)$. Another essential property is that $\ln (a\ast b)=\ln (a)+\ln (b)$, which helps when dealing with products within an expression. Leveraging these properties allows for the transformation of an exponential equation into a form that can be solved with basic algebra.

Isolating Exponential Terms

Isolating the exponential term is a strategic move when solving exponential equations. The goal is to have the term with the variable exponent by itself on one side of the equation so that logarithmic functions can be applied effectively.

The process of isolation involves performing algebraic operations such as addition, subtraction, multiplication, and division. In our example, we isolated the term $e^{x/2}$ by dividing both sides of the equation by -20 after expanding and simplifying the initial equation. Once the term is isolated, as in $e^{x/2}=75$, we apply logarithms to both sides to bring down the exponent. This critical step transforms the equation from an exponential form to a more familiar linear form, paving the way to find the solution for the variable.