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The given equation is:

\[
\log_{10} x = 1 - \frac{1}{2} \log_{10} 25 + 3 \log_{10} 2
\]

Let's solve this step by step.

### Step 1: Simplify the logarithmic terms.
Using logarithmic properties, we can simplify some terms:

- \(\log_{10} 25\): Since \(25 = 5^2\), we can express this as:
  \[
  \log_{10} 25 = 2 \log_{10} 5
  \]

- \(\frac{1}{2} \log_{10} 25\): This becomes:
  \[
  \frac{1}{2} \log_{10} 25 = \frac{1}{2} \times 2 \log_{10} 5 = \log_{10} 5
  \]

- \(3 \log_{10} 2\): This is already simplified.

Now the equation looks like:

\[
\log_{10} x = 1 - \log_{10} 5 + 3 \log_{10} 2
\]

### Step 2: Use the property of combining logarithms.
We can express this in terms of a single logarithm:
\[
1 = \log_{10} 10
\]

Thus, the equation becomes:
\[
\log_{10} x = \log_{10} 10 - \log_{10} 5 + \log_{10} 2^3
\]

Since \(\log_{10} 2^3 = \log_{10} 8\), we now have:
\[
\log_{10} x = \log_{10} 10 - \log_{10} 5 + \log_{10} 8
\]

### Step 3: Combine the logarithms.
Using the logarithmic property \(\log_{10} a - \log_{10} b = \log_{10} \left(\frac{a}{b}\right)\), we get:
\[
\log_{10} x = \log_{10} \left(\frac{10 \times 8}{5}\right)
\]

Simplifying the expression inside the logarithm:
\[
\frac{10 \times 8}{5} = \frac{80}{5} = 16
\]

Thus, the equation becomes:
\[
\log_{10} x = \log_{10} 16
\]

### Step 4: Solve for \(x\).
Since the logarithms on both sides are equal, we can conclude:
\[
x = 16
\]

The solution to the equation is \(x = 16\).

Would you like further details or have any questions?

Here are some related questions for further practice:
1. How do logarithmic properties help in simplifying equations?
2. What is the general rule for adding and subtracting logarithms?
3. Can you solve \(\log_{10} (x^2) = \log_{10} (4x)\)?
4. How would you solve \(\log_{10} (3x) = 2 + \log_{10} 5\)?
5. What is the relationship between natural logarithms and common logarithms?

**Tip:** Remember that logarithmic properties like the product, quotient, and power rules are key tools for simplifying complex logarithmic expressions.

Step-by-step solution to the logarithmic equation lg x = 1 - 1/2 lg 25 + 3 lg 2. Explore logarithmic properties, including the product, quotient, and power rules. Suitable for high school students in Grades 9-12.

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