Problem 28 - Algebra 2

Question

Solve each equation. Check your solutions. \(\log _{7} 24-\log _{7}(y+5)=\log _{7} 8\)

Step-by-Step Solution

Answer

The solution is \(y = -2\).

Step 1: Apply the Logarithmic Property

Recall the property of logarithms: \(\log_b M - \log_b N = \log_b \left( \frac{M}{N} \right)\). Apply it to the left side of the given equation: \(\log_7 24 - \log_7 (y+5) = \log_7 \left( \frac{24}{y+5} \right)\). Substitute this into the equation, so it becomes: \(\log_7 \left( \frac{24}{y+5} \right) = \log_7 8\).

Step 2: Eliminate the Logarithms

Since the logarithms are equal, their arguments must also be equal. Therefore, set \(\frac{24}{y+5} = 8\). We can do this because if \(\log_b X = \log_b Y\), then \(X = Y\).

Step 3: Solve the Equation for y

Now, solve the equation \(\frac{24}{y+5} = 8\). Multiply both sides by \(y+5\) to get rid of the fraction: \(24 = 8(y+5)\).

Step 4: Simplify and Solve for y

Distribute the 8 on the right side of the equation: \(24 = 8y + 40\). Then, subtract 40 from both sides to get \(24 - 40 = 8y\). This simplifies to \(-16 = 8y\). Divide both sides by 8 to solve for \(y\): \(y = -2\).

Step 5: Verify the Solution

To check the solution, substitute \(y = -2\) back into the original equation. Calculate \(\log_7 24 - \log_7 (3)\) and verify it equals \(\log_7 8\). Since 24 divided by 3 is 8, \(\log_7 24 - \log_7 3 = \log_7 8\) confirms the solution is correct.

Key Concepts

Logarithmic Properties

Logarithmic properties are very useful tools when dealing with logarithmic equations. They help simplify complex expressions, making solving equations easier. One of the key properties is:

• Difference of logarithms: \[\log_b M - \log_b N = \log_b \left( \frac{M}{N} \right)\]This property states that when you subtract two logarithms with the same base, you can compress the expression into a single log with the division of their arguments.

In our exercise, we used this property to simplify the left-side expression \[\log_7 24 - \log_7 (y+5) \]into \[\log_7 \left( \frac{24}{y+5} \right)\].These insights effectively reduce complicated equations, providing an approachable path for solving them.
Remember, the argument inside the logarithm should remain positive to ensure the log functions are defined.

Logarithmic Equations

Logarithmic equations, such as \[\log_b \left( \frac{24}{y+5} \right) = \log_b 8\],involve equating two logarithms to one another. Identifying and applying properties of logs helps us simplify and solve these equations.To solve logarithmic equations:

• Combine logs using their properties when possible.
• Once in the form \(\log_b X = \log_b Y\), remember that if the logs are equal, their arguments are equal: \(X = Y\).

In our exercise, we simplified the equation and then set the arguments equal to each other:\[\frac{24}{y+5} = 8\]This step allows us to dismiss the logs entirely, transitioning from a log-based problem to a classical algebraic equation easy to solve.
This method is efficient because it leverages the principle that logarithms with the same base and equal arguments must be equal themselves.

Checking Solutions

Checking your solutions in logarithmic equations is essential to ensure their correctness. Incorrect solutions can arise from errors in simplification or arithmetic errors.To verify a solution:

• Plug the value back into the original equation to ensure both sides are equal.
• Make sure the argument of each logarithm remains positive, as logs of non-positive numbers are undefined.

In our exercise, substituting \(y = -2\) back, we compute:\[\log_7 24 - \log_7 3 = \log_7 8\],given \(3\) is positive and derived from \(y+5\).Since both sides equal \(\log_7 8\), the solution is verified. This step is crucial and affirms that the problem-solving process was executed correctly and the solution complies with all conditions of logarithm definitions.