Problem 54 - College Algebra

Question

For the following exercises, use a graphing calculator to find approximate solutions to each equation.$$\log (2 x-3)+2=-\log (2 x-3)+5$$

Step-by-Step Solution

Answer

The approximate solution is \( x \approx 17.31 \).

Step 1: Simplify the Equation

Let's start by simplifying the given equation:\[ \log (2x - 3) + 2 = -\log (2x - 3) + 5. \] Move the logs to one side to isolate them. We have:\[ \log (2x - 3) + \log (2x - 3) = 5 - 2. \] This simplifies to: \[ 2\log (2x - 3) = 3. \]

Step 2: Solve the Logarithmic Equation

Divide both sides by 2 to isolate the logarithm term:\[ \log (2x - 3) = \frac{3}{2}. \] This implies that \( 2x - 3 \) must be such that the logarithm can equal \( \frac{3}{2} \). Convert the logarithmic form to exponential form. Since we know the base is 10, we have:\[ 2x - 3 = 10^{\frac{3}{2}}. \]

Step 3: Solve for \(x\)

Calculate \(10^{\frac{3}{2}}\) which means \(\sqrt{10^3}\) or \(\sqrt{1000}\). Simplifying, we get \(\sqrt{1000} \approx 31.62\). Thus, \[ 2x - 3 = 31.62. \] Add 3 to both sides: \[ 2x = 34.62. \] Finally, divide by 2: \[ x \approx 17.31. \]

Step 4: Verify the Solution Using the Graphing Calculator

To ensure the solution is correct, use a graphing calculator to graph the original equation: \( y = \log (2x-3) + 2 \) and \( y = -\log(2x-3) + 5 \). Find the x-value where these two graphs intersect. This should confirm \( x \approx 17.31 \).

Key Concepts

Logarithmic Functions

Logarithmic functions are mathematical expressions that show how one number can be expressed as a power of another. In simpler terms, if you have a base number that is raised to a certain power to result in another number, the logarithm tells you what that power is. For instance, if we write \( \log_b (x) = y \), it implies \( b^y = x \.\) Here, \( b \) is the base, and \( y \) is the exponent needed to achieve \( x \). In most cases, when we talk about logarithms, the base is 10 (the common logarithm) unless otherwise stated.

Understanding logarithmic functions is crucial when solving logarithmic equations like the one we explored. Such equations often involve manipulating these functions to simplify and solve for the unknown variables.

• Logarithmic functions can be thought of as 'inverse' operations of exponentiation.
• The properties of logarithms, such as the product, quotient, and power rules, are handy tools when solving equations.

By working through these principles, you can transform seemingly complex equations into simpler forms that are more readily solvable.

Exponentiation

Exponentiation is the mathematical operation of raising one number, the base, to the power of another number, the exponent. For example, in the expression \( b^y \), \( b \) is the base and \( y \) is the exponent. This operation is vital in solving logarithmic equations because it allows us to reverse the logarithmic form, helping us find the values of concealed variables.

In the provided exercise, we converted the logarithmic form into an exponential form to isolate and solve for \( x \). This conversion from \( \log (2x - 3) = \frac{3}{2} \) to \( 2x - 3 = 10^{\frac{3}{2}} \) shows how exponentiation is applied to make sense of logarithmic problems.

• Exponentiation can be visualized as repeated multiplication.
• When solving logarithmic equations, moving between logarithmic and exponential forms helps to find solutions.

By understanding exponentiation better, students can effectively tackle logarithmic problems, like the one described, with confidence.

Graphing Calculators

Graphing calculators are powerful tools that help visualize complex equations and graph their solutions. These calculators are particularly helpful in finding where graphs intersect, which corresponds to the solution of equations like the one in the exercise.

When dealing with two functions, \( y = \log (2x-3) + 2 \) and \( y = -\log (2x-3) + 5 \,\) plotting these on a graphing calculator will show where they intersect. This intersection point directly yields the solution of the equation, confirming the analytical solution computed manually.

• Graphing calculators help in visualizing the behavior of logarithms and other functions.
• They are excellent for checking the accuracy of solutions obtained manually.

For students new to this concept, using a graphing calculator is a good way to bridge the gap between theoretical learning and practical problem-solving.

Equation Solving Steps

Solving equations systematically involves following a series of logical steps. This structured approach not only simplifies the task but also minimizes errors, leading to accurate solutions.

For the given logarithmic equation, the solution involved several clear steps: simplifying the expression, converting to exponential form, isolating variables, and finally using graphing technology to verify results.

• Simplifying equations reduces complexity by combining like terms and utilizing properties of logarithms.
• Converting to exponential form helps isolate the variable you're solving for.
• Checking the solution using a graph ensures validation of the result obtained manually.

By adhering to these steps, each part builds on the last, ensuring that no details are overlooked and giving the student a clear path to find the correct solution.