Problem 54 - College Algebra with Corequisite Support

Question

For the following exercises, use a graphing calculator to find approximate solutions to each equation. \(2 \ln (5 x+1)=\frac{1}{2} \ln (-5 x)+1\)

Step-by-Step Solution

Answer

Using a graphing calculator, approximate the solution where the graphs of \(y_1 = (5x+1)^4\) and \(y_2 = e^2(-5x)\) intersect.

Step 1: Simplify the Equation

Reorganize the terms of the equation: \[2 \ln(5x+1) = \frac{1}{2} \ln(-5x) + 1\]This step involves factoring common terms or isolating logarithmic expressions.

Step 2: Apply Logarithmic Properties

Use the properties of logarithms to simplify each logarithmic term. Specifically, multiply through by a common denominator to clear fractions:\[4 \ln(5x+1) = \ln(-5x) + 2\]Here, "\(a \ln(b) = \ln(b^a)\)" has been used.

Step 3: Rewrite in Exponential Form

The equation now contains simplified logarithms:\[\ln((5x+1)^4) = \ln(-5x) + 2\]This can be rewritten as an exponential equation by removing the natural log on both sides.\[(5x+1)^4 = e^{\ln(-5x) + 2}\]

Step 4: Solve the Exponential Equation

Convert the equation from exponential to standard form by expressing both sides in terms of exponentials:\[(5x+1)^4 = e^2(-5x)\]This simplifies further as: \[625x^4 + 500x^3 + 150x^2 + 20x + 1 = -5xe^2\] which is complex to solve algebraically.

Step 5: Use a Graphing Calculator

Using a graphing calculator, input the functions\[y_1 = (5x+1)^4\]and \[y_2 = e^2(-5x)\].Find the intersection points of these graphs.The x-coordinates of the intersections are solutions to the equation.

Key Concepts

Logarithmic Equations

Logarithmic equations involve logarithms, usually as the main operations on the variable. In these types of equations, you are often asked to find the value of the variable that makes the equation true. Logarithms, the inverse functions of exponentials, help solve equations involving exponentiation. Some important properties of logarithms that are often used include:

• Product Property: \(\log_b(MN) = \log_b(M) + \log_b(N)\)
• Quotient Property: \(\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\)
• Power Property: \(a \log_b(M) = \log_b(M^a)\)

Understanding these properties is key to manipulating and solving logarithmic equations. In our exercise, these properties were used to transform and simplify the logarithmic terms, making the equation easier to work with.

Graphing Calculator Usage

Graphing calculators are powerful tools that can handle complex calculations and graph nonlinear equations. To solve equations like in our exercise, graphing calculators can find the intersection of two functions. Here's how they can be used:

• Enter each side of the equation as separate functions in the calculator, such as \(y_1 = (5x+1)^4\)and \(y_2 = e^2(-5x)\).
• Graph the functions on the same set of axes.
• Use the graphing calculator's "intersect" function to find where these graphs cross, providing the \(x\) values that solve the equation.

Graphing calculators can handle visual representations, allowing us to see the solutions more clearly and precisely. This is especially useful in solving equations that are otherwise difficult to simplify algebraically.

Exponential Form

Exponential form deals with representing numbers as powers. In the context of solving logarithmic equations, an understanding of exponential form is crucial. For example, when you have \(\ln(a) = b\), it translates into exponential form as \(a = e^b\).This conversion is essential in solving our example since it moved the logs to an exponent, making the numbers easier to manage:

• The logarithmic equation \(\ln((5x+1)^4) = \ln(-5x) + 2\)was converted to \((5x+1)^4 = e^2(-5x)\)by employing exponential form.
• This transformation allows for further manipulation of the equation into a polynomial form, which graphing calculators can evaluate.

Understanding exponential form not only aids in simplifying equations but also in finding real-life applications, such as modeling growth or decay in natural processes.

Solving Equations Step-by-Step

The step-by-step approach to solving equations ensures a structured pathway to find solutions. In this method:

• Start by simplifying the equation, often moving terms around to make it easier to handle.
• Use algebraic properties tailored to the type of equation, such as logarithmic or exponential, to further simplify.
• Expect to sometimes convert from one form to another, like from logarithmic to exponential, to untangle the equation.
• Finally, evaluate or find the solutions using tools like graphing calculators when necessary.

For our exercise, each step built on the previous one, starting from simplifying the logarithms and moving all the way to using a graphing calculator. Following these steps closely can demystify complex problems and lead to accurate solutions.