Find the value of x, when log3 x + log3 10 = log3 5 + log3 6 - log3 2

To solve for \( x \) in the equation

\[
\log_3 x + \log_3 10 = \log_3 5 + \log_3 6 - \log_3 2,
\]

we can use the properties of logarithms to simplify and solve the equation.

### Step 1: Use the property \(\log_a b + \log_a c = \log_a (bc)\)

First, apply the property to combine the logarithms on each side:

\[
\log_3 (x \cdot 10) = \log_3 (5 \cdot 6) - \log_3 2
\]

This simplifies to:

\[
\log_3 (10x) = \log_3 (30) - \log_3 2
\]

### Step 2: Use the property \(\log_a b - \log_a c = \log_a \left(\frac{b}{c}\right)\)

Now, apply the property to the right side of the equation:

\[
\log_3 (10x) = \log_3 \left(\frac{30}{2}\right)
\]

Simplifying the fraction:

\[
\log_3 (10x) = \log_3 (15)
\]

### Step 3: Since \(\log_3 a = \log_3 b \Rightarrow a = b\)

We now have:

\[
10x = 15
\]

### Step 4: Solve for \( x \)

Divide both sides by 10:

\[
x = \frac{15}{10} = 1.5
\]

So, the value of \( x \) is \( \boxed{1.5} \).

Would you like more details or have any questions? Here are some related problems you might find useful:

1. Solve \( \log_2 (x + 3) + \log_2 (x - 1) = 3 \).
2. Find \( x \) if \( \log_4 x + \log_4 (x - 1) = 1 \).
3. Solve \( \log_5 (2x + 3) = 2 - \log_5 (x - 1) \).
4. Determine \( x \) for \( \log_3 (x^2 - 4) = 2 \).
5. If \( \log_2 (3x + 1) = \log_2 (x + 5) \), find \( x \).

**Tip:** Remember that the properties of logarithms can greatly simplify complex logarithmic equations, making them easier to solve.

This step-by-step guide teaches you how to solve the logarithmic equation log3 x + log3 10 = log3 5 + log3 6 - log3 2. Learn key logarithmic properties and formulas, suitable for high school students in Grades 9-12.

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