We are given the equation:

$3^{\log(x)} + 3x^{\log(3)} = 36$

Step 1: Rewrite the terms

We can simplify the powers using logarithmic and exponential rules:

• $3^{\log(x)}$: We know from logarithmic properties that $a^{\log_b(c)} = c^{\log_b(a)}$, so $3^{\log(x)} = x^{\log(3)}$.

Thus, the equation becomes:

$x^{\log(3)} + 3x^{\log(3)} = 36$

Step 2: Combine like terms

Since both terms involve $x^{\log(3)}$, we can factor it out:

$x^{\log(3)} (1 + 3) = 36$

This simplifies to:

$4x^{\log(3)} = 36$

Step 3: Solve for $x^{\log(3)}$

Now, divide both sides by 4:

$x^{\log(3)} = \frac{36}{4} = 9$

Step 4: Solve for $x$

We know $x^{\log(3)} = 9$, and 9 can be written as $3^2$. So:

$x^{\log(3)} = 3^2$

Now, take the logarithm of both sides:

$\log(3) \cdot \log(x) = 2\log(3)$

Dividing both sides by $\log(3)$:

$\log(x) = 2$

Exponentiate both sides to solve for $x$:

$x = 10^2 = 100$

Final Answer:

$x = 100$

Would you like more details, or do you have any questions?

Here are 5 related questions:

1. How does the logarithmic identity $a^{\log_b(c)} = c^{\log_b(a)}$ work?
2. How do we factor expressions involving powers with logarithms?
3. Can logarithms with different bases be converted to the same base?
4. How does the property of exponents help simplify equations involving logarithms?
5. What are the real-world applications of logarithmic equations?

Tip: When working with logarithmic equations, try to express all terms in a similar base for easier manipulation.