Problem 330 - Precalculus

Question

Use the one-to-one property of logarithms to solve. \(\log _{4}(6-m)=\log _{4} 3 m\)

Step-by-Step Solution

Answer

\( m = \frac{3}{2} \).

Step 1: Apply One-to-One Property

The one-to-one property of logarithms states that if \( \log_b (M) = \log_b (N) \), then \( M = N \). Therefore, from \( \log_{4}(6-m) = \log_{4} 3m \), we can write: \[ 6 - m = 3m \].

Step 2: Solve for m

Starting with the equation from Step 1, \( 6 - m = 3m \), add \( m \) to both sides to get: \[ 6 = 4m \] Now, divide both sides by 4 to solve for \( m \): \[ m = \frac{6}{4} \], which simplifies to \[ m = \frac{3}{2} \].

Key Concepts

One-to-One Property

In logarithmic equations, a useful property is the one-to-one property. This property allows us to equate the arguments of two logarithms directly, given that their bases are the same. If you have

• \( \log_b(M) = \log_b(N) \)

then you can confidently write

• \( M = N \)

This means that the expressions inside the logarithms are equal to each other. This property is incredibly helpful for simplifying and solving logarithmic equations. For example, consider

• \( \log_4(6-m) = \log_4(3m) \)

Using the one-to-one property, we can directly conclude

• \( 6-m = 3m \)

This step simplifies the equation significantly and allows us to focus on solving a basic algebraic equation.

Solving Logarithmic Equations

Once you've used the one-to-one property, you often have a simpler equation to work with. The next step involves solving the resulting algebraic equation. Let's go through the process:
Starting from our specific equation

• \( 6 - m = 3m \)

The first step to solve for \( m \) is to get all terms containing \( m \) on one side of the equation.

• Add \( m \) to both sides:
• \( 6 = 4m \)

Now, divide both sides by 4 to isolate \( m \):

• \( m = \frac{6}{4} \)

Finally, simplify the fraction to get:

• \( m = \frac{3}{2} \)

This process of isolating the variable involves basic algebraic manipulation techniques which are key to solving equations.

Simplifying Fractions

When you solve an equation and end up with a fraction, simplifying it is the next step. Simplifying fractions makes them easier to interpret and use in further calculations. Here’s how to simplify the fraction from our example:

• The fraction we obtained is \( \frac{6}{4} \).
• This can be simplified by dividing both the numerator and denominator by their greatest common divisor (GCD), which in this case is 2.

The steps are as follows:

• Divide 6 by 2 to get 3.
• Divide 4 by 2 to get 2.
• Thus, \( \frac{6}{4} \) simplifies to \( \frac{3}{2} \).

Simplifying fractions ensures that your answers are in their most straightforward and easily understandable form, which is important for both interpreting the result and using it in further mathematical operations.