Problem 21 - Precalculus : Building Concepts and Connections

Question

Solve the exponential equation. Round to three decimal places, when needed. $$1000 e^{0.04 x}=2000$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = \frac{ln(2)}{0.04}\). The exact numerical value would depend on the value of the natural logarithm of 2, and if required, it should be rounded off to three decimal places.

Step 1: Isolate the exponential expression

First thing to do is divide both sides of the equation by 1000, to isolate the term involving exponential function. This gives the equation \(e^{0.04x} = 2\).

Step 2: Apply natural logarithm on both sides

Now, apply natural logarithm (ln) on both sides of the equation. This serves to reduce the exponential format and will yield \(ln( e^{0.04x}) = ln(2)\). Since the natural logarithm and exponential function are inverse operations of each other, we get \(0.04x = ln(2)\).

Step 3: Solve for x

Solve for x by dividing both sides by 0.04. This gives \(x = \frac{ln(2)}{0.04}\).

Key Concepts

Natural Logarithm

A natural logarithm, often denoted as "ln", is a logarithm to the base of the mathematical constant \(e\), which is approximately equal to 2.718. It's called "natural" because of its frequent appearance in natural growth processes, such as population growth or radioactive decay. When you have an equation where something is raised to the power of \(e\), you often use the natural logarithm to "bring down the exponent" and make the equation easier to solve. The natural logarithm is especially useful because it transforms exponential equations into linear equations, where the exponent becomes a coefficient.

If you have \(e^y = x\), you can apply the natural logarithm to both sides to get \(y = ln(x)\). This is helpful in solving for the unknown in the exponent, like in the original exercise with \(e^{0.04x} = 2\). By applying \(ln\) to both sides, you use the properties of logarithms to rewrite the equation in a simpler form.

Solving Equations

Solving equations is a fundamental skill in algebra where the aim is to find the value(s) of the unknown variable(s) that make the equation true. For exponential equations, like the given \(1000 e^{0.04 x} = 2000\), you'll often start by isolating the exponential part. This step involves straightforward operations like addition, subtraction, multiplication, or division to simplify the equation. In this example, dividing both sides by 1000 isolates the term with the exponential, reducing it to \(e^{0.04x} = 2\).

• Always ensure that the exponential part is alone on one side of the equation before applying a logarithm or any operation to both sides.
• Check your results by substituting the solution back into the original equation to ensure it balances.

After isolating the exponential term, applying a logarithm is often the next step if you do not have equal bases. This action makes it easier to solve for the variable in the exponent.

Inverse Operations

Inverse operations are mathematical processes that undo each other, much like addition and subtraction or multiplication and division. In the context of solving exponential equations, the natural logarithm and the exponential function are inverse operations. This means that \(ln(e^y)\) results in \(y\). This property is crucial when solving for a variable in an exponential equation because it allows the exponent to be extracted and resolved more easily.

When you initially isolate \(e^{0.04x}\) on one side of the equation, and you apply \(ln\) to both sides, it makes use of the inverse relationship to simplify the expression to \(0.04x = ln(2)\).

• Inverse operations allow us to "reverse" the effect of the original operation, bringing complex expressions into more manageable forms.
• Understanding inverse operations is key to solving many algebraic equations, especially those involving exponential and logarithmic functions.

By executing the inverse operation, you unlock the means to straightforwardly solve for the unknown variable in the equation.