Problem 25 - College Algebra with Corequisite Support

Question

For the following exercises, use logarithms to solve. \(8 e^{-5 x-2}-4=-90\)

Step-by-Step Solution

Answer

The exponential equation cannot be solved with a negative on the right side. Re-evaluate the problem context.

Step 1: Isolate the exponential expression

Start by adding 4 to both sides of the equation to isolate the exponential expression.\[ 8e^{-5x-2} = -90 + 4 \]Simplify the right side:\[ 8e^{-5x-2} = -86 \]

Step 2: Solve for the exponential term

Divide both sides by 8 to solve for the exponential term:\[ e^{-5x-2} = \frac{-86}{8} \]This simplifies to:\[ e^{-5x-2} = -10.75 \]Realize there is an issue here, as the exponential term cannot be negative. Re-examine any mistakes in simplification.

Step 3: Correction - Recalculate properly

Upon further consideration, re-evaluate the set-up of the equation, ensuring correct calculations initially. Revisit step assumptions as there should not be negative values for exponential terms, indicating a potential mistake in original problem setting or solution path. Consult for correct options.

Step 4: Numerical Property Check

Recall that exponential functions, such as those involving \(e^x\), are always positive, suggesting an issue with the provided equation context. Return to verify numerical setups since negative results cannot be directly solved in traditional perspective without lateral consideration like complex numbers if applicable.

Key Concepts

Logarithms

Logarithms are essential tools in mathematics, often used to solve exponential equations. They allow us to work with exponential terms more easily by transforming them into manageable linear forms. To use logarithms in solving exponential equations, we typically apply the natural logarithm (ln) if the base of the exponent is the natural number, e, which is approximately 2.718.

Logarithms have some key properties:

• The logarithm of a product is the sum of the logarithms: \(\log_b(xy) = \log_b(x) + \log_b(y)\)
• The logarithm of a quotient is the difference of the logarithms: \(\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\)
• The logarithm of a power is the exponent times the logarithm: \(\log_b(x^y) = y \cdot \log_b(x)\)

To solve exponential equations like \(e^{-5x-2} = -10.75\), you would typically take the natural logarithm of both sides. However, as noted, there is an issue with the context since exponential equations with a base of e cannot equal a negative number in real numbers. In such cases, examining the problem setup or exploring complex numbers is necessary.

Exponential Functions

Exponential functions are a class of functions defined as \(f(x) = a \cdot b^x\), where a is a constant and b is the base of the exponential function, typically greater than 0. They are characterized by a constant rate of growth or decay. The most common base for exponential functions in calculus and real-world applications is the natural number, e.

Exponential functions have several key properties:

• The function \(e^x\) is always positive for any real number x, meaning it can never be negative.
• The rate of change of the function is proportional to its current value, as given by the derivative \(\frac{d}{dx} \left(e^x\right) = e^x\).
• Exponential functions grow faster than any polynomial function as x approaches infinity.

When attempting to solve equations involving exponential functions where the result is negative, it's crucial to properly examine the context and ensure the equation's validity in the real number system.

Complex Numbers

Complex numbers extend the concept of one-dimensional number line operations to two-dimensional number planes, allowing the solutions of equations that have no real solutions. A complex number is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit that satisfies \(i^2 = -1\).

In the context of exponential equations, if a given solution suggests negatives that are unresolvable within the real number system, it's vital to evaluate whether complex solutions might be applicable. By utilizing Euler's formula, \(e^{ix} = \cos(x) + i\sin(x)\), we can further explore solutions:

• Complex solutions can offer valid results for negative outcomes in exponential equations through imaginative exploration.
• They help explain cases where traditional approaches imply errors, such as negative exponential expressions.
• Understanding the role of complex numbers requires a shift from familiar real-number thinking to a broader and more flexible framework.

This framework can reveal insight into previously unsolvable problems or those perceived as erroneous due to limitations of real number applicability.