Problem 31 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each exponential equation by taking the logarithm on both sides. Express the solution set in terms of logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$1250 e^{0.055 x}=3750$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 20.09\).

Step 1: Simplify the equation

The first step is to simplify the equation by dividing both sides by 1250. This gives the equation \( e^{0.055 x} = 3 \)

Step 2: Apply logarithms

Next, natural logarithms should be applied to both sides of the equation to remove the exponential term, resulting in the equation \(0.055x = \ln(3)\)

Step 3: Solve for x

Finally, the last step is to solve for x, which can be done by dividing both sides of the equation by 0.055. This gives \(x = \frac{\ln(3)}{0.055}\)

Step 4: Calculate the value of x

Last, to solve for x numerically, the natural logarithm of 3 is divided by 0.055 using a calculator, giving the solution \(x = 20.09\)

Key Concepts

Logarithms

Logarithms are an essential mathematical tool used to work with exponential expressions. Think of them as the inverse operations of exponents. When you raise a base to a power, you get a number. A logarithm tells you the power to which the base needs to be raised to achieve that number.

For example, in the equation \( b^y = x \), the logarithm of \( x \) with base \( b \) is \( y \). This is written as \( y = \log_b(x) \). In simpler terms, logarithms help us work backwards from the result of an exponentiation to the exponent itself.

In solving exponential equations like \( 1250 e^{0.055x} = 3750 \), logarithms allow us to "undo" the exponentiation by transforming the exponential equation into a linear one. This is done by taking the logarithm of both sides, making it easier to isolate and solve for the variable.

Natural Logarithm

The natural logarithm, often denoted as \( \ln \), specifically uses the base \( e \), a special mathematical constant approximately equal to 2.718. Natural logarithms are prevalent in mathematics due to their simplicity and the properties of the number \( e \).

When using natural logarithms, the expression \( \ln(x) \) indicates the power to which \( e \) must be raised to result in \( x \). This means if \( e^y = x \), then \( y = \ln(x) \).

For the equation \( e^{0.055x} = 3 \), applying \( \ln \) to both sides helps to cancel out the exponential function, simplifying our equation to \( 0.055x = \ln(3) \). From here, finding \( x \) becomes straightforward as the equation is now linear.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. In the context of exponential equations like \( e^{0.055x} = 3 \), logarithms play a crucial role.

Here, once the equation is transformed into \( 0.055x = \ln(3) \) using logarithms, we have a linear equation. The next step is to isolate the variable \( x \).

• Divide both sides by the coefficient of \( x \) (in this case, 0.055).
• This yields \( x = \frac{\ln(3)}{0.055} \).

By calculating this expression, we find that \( x \approx 20.09 \), a numerical solution rounded to two decimal places. Breaking complex equations into simpler elements through these methods makes it easier to understand and solve them accurately.