Problem 18 - College Algebra with Corequisite Support

Question

For the following exercises, use logarithms to solve. \(-5 e^{9 x-8}-8=-62\)

Step-by-Step Solution

Answer

\(x = \frac{\ln(10.8) + 8}{9}\).

Step 1: Isolate the Exponential Expression

Start by isolating the exponential expression on one side of the equation. We have: \(-5 e^{9x-8} - 8 = -62\).Add 8 to both sides:\(-5 e^{9x-8} = -54\).Divide both sides by -5:\(e^{9x-8} = 10.8\).

Step 2: Apply the Natural Logarithm

To eliminate the exponential function and solve for \(x\), take the natural logarithm of both sides:\(\ln(e^{9x-8}) = \ln(10.8)\).Using the logarithmic identity \(\ln(e^y) = y\), we get:\(9x - 8 = \ln(10.8)\).

Step 3: Solve for x

Now, solve for \(x\) by first adding 8 to both sides:\(9x = \ln(10.8) + 8\).Next, divide by 9 to isolate \(x\):\(x = \frac{\ln(10.8) + 8}{9}\).

Key Concepts

Exponential Equations

Understanding exponential equations is essential in mathematics. They frequently appear across different domains, particularly in natural sciences and finance. An exponential equation includes a variable as part of the exponent, like this one: \( e^{9x-8} \). In solving such equations, the goal is to get the variable (here \( x \)) out of the exponent.

There are some handy steps to follow:

• Identify and isolate the exponential expression. This means rearranging the equation so that the exponential component stands alone on one side.
• Once isolated, use logarithms to "break down" the expression, making it more manageable and solvable for \( x \).
• Finally, use algebraic techniques to solve for the variable.

A systematic approach makes solving exponential equations easier. You'll know you're on the right track when the variable moves from being an exponent to a regular term within the equation.

Natural Logarithms

Natural logarithms are a specific kind of logarithm with the base \( e \), where \( e \approx 2.718 \.\). They serve an important function in solving equations where the base is \( e \). For an equation like \( e^{9x-8} = 10.8 \), applying a natural logarithm transforms it into a more straightforward algebraic equation.

Here's how it works:

• When you apply \( \ln \) to each side of the equation, the left side becomes \( \ln(e^{9x-8}) \).
• The logarithm and the exponential function with the same base are inverse operations, simplifying the expression on the left side to \( 9x-8 \).
• The equation \( \ln(e^{9x-8}) = \ln(10.8) \) now looks much simpler, making it possible to solve for \( x \) more easily.

Natural logarithms are robust tools, particularly valuable in transforming complex exponential equations into something manageable by standard algebraic methods.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging equations to make them simpler and solvable for unknown variables like \( x \.\). After applying the natural logarithm, we use algebraic manipulation to isolate \( x \.\).

Here's how the manipulation unfolds:

• You start by having \( 9x - 8 = \ln(10.8) \) after logging both sides of the equation.
• Add 8 to both sides to isolate the term with \( x \), resulting in \( 9x = \ln(10.8) + 8 \).
• Divide everything by 9 to solve for \( x \), giving \( x = \frac{\ln(10.8) + 8}{9} \).

Such algebraic steps ensure precision and ease in solving equations. The clear step-by-step approach ensures that each variable is systematically addressed.