Problem 49 - Beginning and Intermediate Algebra

Question

Solve the logarithmic equation algebraically. Then check using a graphing calculator. $$\log (x+8)-\log (x+1)=\log 6$$

Step-by-Step Solution

Answer

The algebraic solution for the logarithmic equation \(\log (x+8)-\log (x+1)=\log 6\) is \(x = \frac{2}{5}\). This was confirmed using a graphing calculator by comparing the values of \(y_1 = \log (x+8)-\log (x+1)\) and \(y_2 = \log 6\) at \(x = \frac{2}{5}\), both yielding a value of approximately \(0.77815125\).

Step 1: Apply the properties of logarithms

Our given equation is: \[\log (x+8)-\log (x+1)=\log 6\] We can apply the properties of logarithms, specifically the quotient rule which states, \(\log_a x - \log_a y = \log_a \left(\frac{x}{y}\right)\), and rewrite the equation as: \[\log \frac{(x+8)}{(x+1)} = \log 6\]

Step 2: Remove the logarithms from both sides

Since both sides of the equation have a logarithm with the same base, we can remove the logarithms by setting the inside expressions equal to each other: \[\frac{(x+8)}{(x+1)} = 6\]

Step 3: Solve for x

Next, we'll solve for x by setting up a cross-multiplication and simplifying the equation. 1. Cross-multiply: \[(x+8) = 6(x+1)\] 2. Distribute 6 to terms inside the parentheses: \[(x+8) = 6x + 6\] 3. Subtract x from both sides: \[8 = 5x + 6\] 4. Subtract 6 from both sides: \[2 = 5x\] 5. Divide both sides by 5: \[x = \frac{2}{5}\] The algebraic solution for x is \(\frac{2}{5}\).

Step 4: Check the solution using a graphing calculator

Now we need to check whether the solution found is correct using a graphing calculator. To do this, we need to graph both the left and right sides of the equation: 1. Left side: \(y_1 = \log (x+8) - \log (x+1)\) 2. Right side: \(y_2 = \log 6\) Next, we'll plug in our solution, \(\frac{2}{5}\), into the graphing calculator for x and compare the values of y for both equations: \(y_1(\frac{2}{5}) = \log (\frac{2}{5}+8)-\log (\frac{2}{5}+1)\) \(y_1(\frac{2}{5}) \approx 0.77815125\) \(y_2(\frac{2}{5}) = \log 6\) \(y_2(\frac{2}{5}) \approx 0.77815125\) As we can see, the values of y for both the left and right sides of the equation are approximately equal, indicating that our solution, \(x = \frac{2}{5}\), is correct.

Key Concepts

Properties of Logarithms

When solving logarithmic equations, understanding the properties of logarithms is crucial. In our original exercise, we use one of these properties known as the quotient rule. The quotient rule helps us condense the difference of two logarithms into a single logarithm, making the equation simpler and easier to solve.

Here's how the quotient rule works:

• It states that the difference between two logarithms with the same base is equal to the logarithm of the quotient of their arguments.
• In mathematical terms, this is: \[\log_a x - \log_a y = \log_a \left(\frac{x}{y}\right) \]

In the context of our logarithmic equation \(\log (x+8) - \log (x+1)\), applying the quotient rule simplifies it to:\[\log \frac{(x+8)}{(x+1)} = \log 6 \]
Once simplified, the logarithms can be dropped, allowing us to work with a straightforward algebraic equation. Properties like the quotient rule are fundamental tools for manipulating and solving logarithmic equations efficiently.

Algebraic Solutions

After using the properties of logarithms to simplify a logarithmic equation, the next step is to solve the resulting equation algebraically. This often involves removing the logarithms by equating the inner expressions, which brings us to simpler algebraic equations.

In our exercise, we reached the equation:\[\frac{(x+8)}{(x+1)} = 6\]From here, algebraic steps include:

• Cross-multiplying to eliminate the fraction, resulting in:\[(x+8) = 6(x+1)\]
• Distributing and rearranging terms to isolate \(x\), yielding:\[8 = 6x + 6 - x\]
• Simplifying further to find \(x\):\[2 = 5x\]
• Finally, solving for \(x\) by dividing both sides:\[x = \frac{2}{5}\]

The algebraic solution \(x = \frac{2}{5}\) is valid as long as it checks out when substituted back into the original logarithmic expression, keeping in mind any restrictions in the domain due to the logarithm functions involved.

Graphing Calculators

A graphing calculator serves as a powerful tool in verifying the solutions of mathematical equations, including logarithmic equations. To ascertain the correctness of our solution, we employ a graphing calculator to compare the values of the two sides of our equation.

In practical terms:

• We graph the left-hand side function: \[y_1 = \log (x+8) - \log (x+1)\]
• We graph the constant right-hand side: \[y_2 = \log 6\]

Subsequently, we input our solution \(x = \frac{2}{5}\) into the calculator and observe:

• The value of \(y_1\) at \(x = \frac{2}{5}\) is approximately 0.778, matching \(y_2\).

This confirms that our algebraic solution is correct. Graphing calculators are especially useful as they provide a visual representation and immediate comparison, validating complex solutions effortlessly. Additionally, they can illustrate potential discrepancies or confirm domain restrictions inherent to logarithmic functions.