Problem 62 - Beginning and Intermediate Algebra

Question

Solve the logarithmic equation algebraically. Then check using a graphing calculator. $$2 \log 50=3 \log 25+\log (x-2)$$

Step-by-Step Solution

Answer

The short answer to solving the logarithmic equation algebraically is: 1. Combine the logarithms into a single logarithm \( \log{\frac{2500}{15625}} = \log(x-2)\). 2. Since both logarithms are equal, their arguments must be equal \(\frac{2500}{15625} = x-2\). 3. Solve for x: \(x = \frac{5}{12}\). To check the solution graphically, use a graphing calculator to plot the equation \(2 \log 50 - 3 \log 25 - \log(x-2) = 0\) and look for the x-intercept at \(x = \frac{5}{12}\). If the graph intersects the x-axis at this point, the solution is correct.

Step 1: Combine logarithms

Notice that we can rewrite the equation using the product rule "reverse" of logarithm: \(2 \log 50 - 3 \log 25 = \log(x - 2)\). Now we should use the properties of the logarithm to simplify the expressions further.

Step 2: Apply properties of logarithms

The left side of the equation can be simplified using the power rule of logarithms: \(\log 50^2 - \log 25^3 = \log(x-2)\). Simplify the exponents: \(\log 2500 - \log 15625 = \log(x-2)\). Now, we can use the division rule of logarithms: \(\log{\frac{2500}{15625}} = \log(x-2)\).

Step 3: Solve for x

Since both sides of the equation are equal, their logarithm arguments must be equal as well: \(\frac{2500}{15625} = x-2\). Solve for x: \(x -2 = \frac{2500}{15625}\) \(x = \frac{2500}{15625}+2\) \[x = \frac{2500}{6250}\] Finally, simplify the fraction: \(x = \frac{5}{12}\).

Step 4: Check the solution using a graphing calculator

To check the solution graphically, use a graphing calculator to plot the equation: \(2 \log 50 - 3 \log 25 - \log(x-2) = 0\). Look for the x-intercept (the point where the graph crosses the x-axis), which is the solution of the equation. The x-intercept should be \(x = \frac{5}{12}\). If the graph intersects the x-axis at this point, then the solution is correct.

Key Concepts

Properties of Logarithms

Logarithms have several important properties that help in simplifying and solving equations. These properties include:

• Product Rule: This rule states that the logarithm of a product is equal to the sum of the logarithms of the factors. Mathematically, it is expressed as: \[ \ \log (ab) = \log a + \log b \ \]
• Quotient Rule: For the logarithm of a quotient, we can write: \[ \ \log \left(\frac{a}{b}\right) = \log a - \log b \ \]Here, the logarithm of the division of two numbers equals the difference of their logarithms.
• Power Rule: When dealing with exponents inside a logarithm, you can "bring down" the exponent in front of the logarithm: \[ \ \log (a^b) = b \log a \ \]This rule is particularly helpful in simplifying equations with exponents contained within a logarithm.

Applying these properties allows us to transform and simplify complex logarithmic expressions, which is crucial for solving equations algebraically. When you encounter a logarithmic equation, it's helpful to look for opportunities to use these properties to make your work more manageable.

Solving Equations Algebraically

To solve a logarithmic equation algebraically, you use properties of logarithms to simplify and manipulate the equation. The goal is to isolate the variable so you can find its value. Here's a step-by-step process:Begin by applying the Power Rule:

• Identify any constants that can be moved as exponents inside the logarithm.
• For example, in the equation: \(2 \log 50 = 3 \log 25 + \log (x-2)\), we rewrite it as: \(\log 50^2 = \log 25^3 + \log (x-2)\).

Next, use the Quotient Rule to combine the logarithmic terms:

• Simplify into a single log expression where possible by dividing or subtracting: \(\log (50^2) - \log (25^3) = \log (x-2)\).
• It results in \(\log \left(\frac{2500}{15625}\right) = \log (x-2)\).

Now, equate the arguments of the logarithms since they have the same base:

• \(\frac{2500}{15625} = x-2\).
• Solving for \(x\) gives \(x = \frac{5}{12}\).

After finding a solution, it's a good practice to verify it by plugging it back into the original equation or using another method, such as graphing.

Graphing Calculator

A graphing calculator is a fantastic tool to verify solutions of algebraic and logarithmic equations visually. Here's how it assists:

• Plotting the Equation: Enter the simplified form of the equation into the calculator. For the exercise, that would look like: \[ \ 2 \log 50 - 3 \log 25 - \log(x-2) = 0 \ \]
• Finding the X-Intercept: The x-intercept of the graph, where it crosses the x-axis, represents the solution of the equation.
• Check if the point of intersection corresponds to the solution found algebraically. If \(x = \frac{5}{12}\) is the x-intercept, then your algebraic method is validated.

Besides checking work, a graphing calculator helps in understanding how functions behave visually. It can show how subtle changes in the equation affect the graph, thereby deepening comprehension and enhancing problem-solving skills.