You provided the equation:

\[
\log B = 3(\log x - 1) - 2(1 - \log y)
\]

Let's simplify and solve for \( B \) if possible.

### Step 1: Expand the terms inside the parentheses.

\[
\log B = 3\log x - 3 - 2(1 - \log y)
\]

Distribute the \( -2 \):

\[
\log B = 3\log x - 3 - 2 + 2\log y
\]

### Step 2: Combine like terms.

\[
\log B = 3\log x + 2\log y - 5
\]

### Step 3: Express \( B \) in terms of \( x \) and \( y \).

Using properties of logarithms, we can combine the terms on the right-hand side:

\[
\log B = \log(x^3) + \log(y^2) - 5
\]

This simplifies further to:

\[
\log B = \log(x^3 y^2) - 5
\]

Now, use the property \( \log a - b = \log\left(\frac{a}{10^b}\right) \):

\[
\log B = \log\left(\frac{x^3 y^2}{10^5}\right)
\]

Thus:

\[
B = \frac{x^3 y^2}{10^5}
\]

### Final Answer:

\[
B = \frac{x^3 y^2}{100000}
\]

Let me know if you'd like more details or have any questions! Here are some related questions:

1. How can logarithmic properties be used to simplify expressions?
2. What are common mistakes when expanding logarithmic terms?
3. Can this method apply to any logarithmic equation involving multiple variables?
4. How would this equation change if the base of the logarithm were different?
5. What if the equation had a natural logarithm instead?

**Tip:** Always simplify logarithmic expressions by applying the basic logarithm rules: product, quotient, and power properties.

Learn how to simplify the logarithmic equation LogB = 3(logx - 1) - 2(1 - logy) step-by-step, covering logarithmic properties, expansion, and solving for B. This guide is ideal for high school students (Grades 10-12) tackling logarithmic expressions.

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