Problem 44 - College Algebra

Question

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

Step-by-Step Solution

Answer

The solution is \( x = \frac{12}{11} \).

Step 1: Apply Logarithm Property

Use the property of logarithms that allows you to combine two logarithms on the right side of the equation: \[ \log(x) + \log(12) = \log(x \cdot 12) = \log(12x) \] This simplifies the equation to: \[ \log(x+12) = \log(12x) \]

Step 2: Utilize One-to-One Property of Logarithms

Since the logarithms on both sides of the equation have the same base, we can set their arguments equal to each other. This gives: \[ x + 12 = 12x \]

Step 3: Solve the Linear Equation

Rearrange the equation from Step 2 to solve for \(x\):1. Subtract \(x\) from both sides: \[ 12 = 11x \]2. Divide both sides by 11: \[ x = \frac{12}{11} \]

Key Concepts

One-to-One Property

The one-to-one property is a valuable principle when working with logarithmic equations. This property states that if the values of two logarithms with the same base are equal, then their arguments must also be equal.

In simpler terms, if you have an equation of the form \( \log_a(M) = \log_a(N) \), you can deduce that \( M = N \). This is because a logarithmic function is a one-to-one function, meaning each input (argument) corresponds to a unique output (log value).

When solving logarithmic equations, the one-to-one property allows us to eliminate the logarithms by equating the expressions inside the logs. For example, in the equation \( \log(x+12) = \log(12x) \), we apply this property to conclude that \( x+12 = 12x \). This simplification is critical for reducing complex equations into ones that are more straightforward to solve.

Logarithm Properties

Logarithm properties are rules that describe how logarithms behave and interact. These fundamental rules help to simplify logarithmic expressions. Here are some essential properties that are often used:

• Product Rule: \( \log_b(M) + \log_b(N) = \log_b(M \cdot N) \). This means you can combine two logarithms of the same base into a single log that has the product of their arguments.
• Quotient Rule: \( \log_b(M) - \log_b(N) = \log_b\left(\frac{M}{N}\right) \). This property allows you to express the difference of logs as a single log with a divided argument.
• Power Rule: \( \log_b(M^n) = n \cdot \log_b(M) \). This rule states that you can bring a power inside the log out as a coefficient.

In the original problem, the product rule is used. \( \log(x) + \log(12) \) combines to \( \log(12x) \). Understanding and using these properties efficiently is key to solving logarithmic equations.

Solving Linear Equations

Linear equations are algebraic expressions of the form \( ax + b = c \). Solving them involves finding the value of the variable that makes the equation true.

To solve, follow these general steps:

• Isolate: Begin by arranging terms such that the variable term is on one side of the equation and constants are on the other.
• Simplify: Combine like terms if needed and adjust the equation to start simplifying.
• Solve: Divide or multiply to solve for the variable.

In the original exercise, after using the one-to-one property, you have \( x + 12 = 12x \). To solve, subtract \( x \) from both sides to get \( 12 = 11x \). Then, divide by 11 to find \( x = \frac{12}{11} \). Linear equations require careful attention to operations to isolate the variable successfully.