Problem 124 - Elementary and Intermediate Algebra

Question

If \(\log _{b} 3 x=1+\log _{b} x,\) find \(b\).

Step-by-Step Solution

Answer

The value of \( b \) is 3.

Step 1: Set Up the Equation

You start with the equation \( \log_b(3x) = 1 + \log_b(x) \). Both sides of the equation involve the same base \( b \), so we will aim to simplify this.

Step 2: Expand the Logarithm on the Left

Apply the logarithm product rule to the left side: \( \log_b(3x) = \log_b(3) + \log_b(x) \). Now the equation becomes \( \log_b(3) + \log_b(x) = 1 + \log_b(x) \).

Step 3: Eliminate \(\log_b(x)\)

Subtract \( \log_b(x) \) from both sides of the equation: \( \log_b(3) = 1 \).

Step 4: Convert to Exponential Form

Convert \( \log_b(3) = 1 \) to its exponential form: \( 3 = b^1 \), or simply \( 3 = b \).

Step 5: Conclusion

Based on the steps, the value of \( b \) that satisfies the equation is \( 3 \).

Key Concepts

Logarithm Product Rule

Understanding the concept of the logarithm product rule is crucial when dealing with logarithmic equations. This rule states that the logarithm of a product is the sum of the logarithms of the factors. Mathematically, if you have

• \( \log_b(MN) \),

then it can be expanded to:

• \( \log_b(M) + \log_b(N) \).

This means that if you are working with a logarithm that has a product within it, you can separate each factor as an individual logarithm with the same base, and then add them together.
This property directly applies to the step in the provided solution where \( \log_b(3x) \) was expanded to \( \log_b(3) + \log_b(x) \). Applying this rule helps in simplifying and solving logarithmic equations effectively, making complex expressions more manageable.

Exponential Form

Converting logarithmic equations to their exponential form is a powerful technique. It provides a more straightforward approach to finding solutions, especially in solving for unknown bases. The relationship between logarithms and exponents is given by:

• If \( \log_b(a) = c \), then in exponential form it's \( b^c = a \).

Introducing the exponential form transforms the logical approach from adding and subtracting into multiplication and division, which some might find easier to handle intuitively.
In our solution, converting \( \log_b(3) = 1 \) into its exponential form resulted in \( 3 = b^1 \).
By expressing the equation in exponential form, the problem of finding the base simplifies significantly, as shown by the quick determination that \( b = 3 \).
Understanding this conversion helps eliminate logarithms to solve equations where other methods might be overly complex or not apparent.

Solving for Base of Logarithm

Determining the base of a logarithm is often a key step in solving logarithmic equations. After simplifying the initial equation and arriving at an expression such as \( \log_b(a) = c \), converting to the exponential form \( b^c = a \) can be crucial. With exponential form established, you solve directly for the base 'b'.
For instance, given \( \log_b(3) = 1 \), converting it gives \( 3 = b \).
This showcases a simplified path to identifying the base of the logarithm. Always remember:

• The goal is to isolate the base 'b'.
• Use properties of exponents and logarithms creatively to streamline the process.

For students, mastering this technique can demystify seemingly complex logarithmic problems and provide a clear route from problem to solution.