$$ \log\left(10x\right)=\log11 $$$$ +3\log3 $$

We are given the equation:

\[
\log\left(10x\right) = \log 11 + 3\log 3
\]

### Step 1: Simplify the right side
Using the **logarithmic property** $ a \log b = \log(b^a) $, the term $ 3\log 3 $ becomes:

\[
3\log 3 = \log(3^3) = \log(27)
\]

Thus, the equation becomes:

\[
\log(10x) = \log(11) + \log(27)
\]

### Step 2: Use the sum of logarithms
Using the **logarithmic property** $ \log a + \log b = \log(ab) $, the right side simplifies to:

\[
\log(11) + \log(27) = \log(11 \times 27) = \log(297)
\]

### Step 3: Equate the logarithmic expressions
Since both sides of the equation are logarithms, we can equate the arguments:

\[
10x = 297
\]

### Step 4: Solve for $x$
\[
x = \frac{297}{10} = 29.7
\]

### Final Answer
The solution to the equation is:

\[
x = 29.7
\]

---

Do you want further details or have any questions?

Here are 5 related questions to expand your understanding:
1. What are the fundamental properties of logarithms?
2. How do you solve equations involving logarithms with different bases?
3. What happens if logarithms appear on both sides of an equation with different arguments?
4. Can you solve this equation using natural logarithms instead?
5. How does logarithmic differentiation work in calculus?

**Tip:** Always check if the argument inside the logarithm is valid (i.e., positive) before solving.

$$ \log\left(10x\right) = \log 11 + 3\log 3 $$

Explore a step-by-step solution to the logarithmic equation \log(10x) = \log 11 + 3\log 3. This solution covers essential logarithmic properties, including power and sum rules, and guides you to solve for x accurately. Suitable for high school students in Grades 10-12.

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