Problem 44 - College Algebra with Corequisite Support

Question

For the following exercises, solve each equation for \(x\). \(\log (x+12)=\log (x)+\log (12)\)

Step-by-Step Solution

Answer

\( x = \frac{12}{11} \)

Step 1: Understand the Problem

We need to solve the equation \( \log (x+12) = \log (x) + \log (12) \) for \( x \). Both sides of the equation have logarithmic expressions.

Step 2: Use Properties of Logarithms

Start by using the property of logarithms that states \( \log(a) + \log(b) = \log(ab) \). Apply it to the right-hand side: \( \log(x) + \log(12) = \log(12x) \). This gives us the equation: \( \log(x+12) = \log(12x) \).

Step 3: Apply the One-to-One Property of Logarithms

Since the logarithms on both sides have the same base, we can equate their arguments: \( x+12 = 12x \). This is because if \( \log(a) = \log(b) \), then \( a = b \).

Step 4: Solve the Linear Equation

Rearrange the equation \( x+12 = 12x \) to solve for \( x \). Subtract \( x \) from both sides: \( 12 = 11x \). Divide both sides by 11: \( x = \frac{12}{11} \).

Step 5: Verify the Solution

Substitute \( x = \frac{12}{11} \) back into the original equation to verify. Calculate \( \log \left( \frac{12}{11} + 12 \right) = \log \left( \frac{12}{11} \right) + \log(12) \), which holds true as both sides simplify to the same value. This confirms the solution is correct.

Key Concepts

Properties of Logarithms

Logarithms have several important properties that are essential for solving logarithmic equations. One such property is the Product Property of Logarithms. This property states that the logarithm of a product is equal to the sum of the logarithms of its factors. In mathematical terms, this can be expressed as:

• \( \log(a) + \log(b) = \log(ab) \)

In our exercise, we applied this property to the right-hand side of the equation \( \log(x) + \log(12) \), combining it into \( \log(12x) \). By utilizing this property, we simplified the equation into a form that was easier to solve.
Remember, the key is recognizing when you can apply this property to consolidate logarithmic terms, which is a significant step in simplifying and solving equations involving logarithms.

One-to-One Property of Logarithms

The One-to-One Property of Logarithms is another crucial concept for solving equations where logarithmic expressions appear. This property states that if the logarithms of two values are equal, then the values themselves must be equal, assuming the base of the logarithms is the same. Mathematically, it can be described as:

• If \( \log_{b}(a) = \log_{b}(c) \), then \( a = c \)

In our solution, after using the product property to get \( \log(x+12) = \log(12x) \), we used the One-to-One Property to deduce that:

• \( x + 12 = 12x \)

This property allows us to move from the logarithmic domain to a simpler arithmetic one, providing us with a straightforward path to solve for \( x \). It's vital to ensure that the arguments (the values inside the logarithm) fall within the domain of logarithmic functions (i.e., positive numbers) when employing this property.

Linear Equations

Once we applied the One-to-One Property of Logarithms, we were left with a simple linear equation: \( x + 12 = 12x \). Linear equations involve basic algebraic expressions where the variable is raised to the power of one and can generally be solved in a step-by-step manner.
To solve the equation \( x + 12 = 12x \), follow these steps:

• Subtract \( x \) from both sides to isolate terms involving the variable on one side: \( 12 = 11x \).
• Divide both sides of the equation by 11 to solve for \( x \): \( x = \frac{12}{11} \).

Linear equations like this are straightforward once simplified from their original form, and they illustrate the importance of moving terms around to isolate the variable.

Verification of Solutions

After arriving at a solution, it's essential to verify that it satisfies the original equation to ensure correctness. Verification eliminates errors and confirms that the solution lies within the permissible domain of the function.
For the equation \( \log (x+12) = \log (x) + \log (12) \), with the calculated solution \( x = \frac{12}{11} \), check the work by substituting back:

• Calculate \( \log \left( \frac{12}{11} + 12 \right) \).
• Calculate \( \log \left( \frac{12}{11} \right) + \log(12) \).

Both sides should result in equal values, confirming the solution fits the initial equation. Verification not only checks arithmetic correctness but also ensures all steps followed conformed to mathematical rules, such as domain restrictions in logarithms.