\sqrt[5]{\log _3\left(x\right)}+\log _3\left(\sqrt{x}\right)=12

We are tasked with solving the equation:

\[
\sqrt[5]{\log_3(x)} + \log_3\left(\sqrt{x}\right) = 12
\]

### Step 1: Simplify \(\log_3(\sqrt{x})\)

We can rewrite \(\log_3\left(\sqrt{x}\right)\) using logarithmic properties. Recall that \(\log_b(a^n) = n \log_b(a)\), so:

\[
\log_3\left(\sqrt{x}\right) = \log_3\left(x^{1/2}\right) = \frac{1}{2} \log_3(x)
\]

Thus, the equation becomes:

\[
\sqrt[5]{\log_3(x)} + \frac{1}{2} \log_3(x) = 12
\]

### Step 2: Let \( y = \log_3(x) \)

To simplify further, let's set \( y = \log_3(x) \). The equation now becomes:

\[
\sqrt[5]{y} + \frac{1}{2}y = 12
\]

### Step 3: Introduce a substitution for \( \sqrt[5]{y} \)

Let \( z = \sqrt[5]{y} \), which means \( z^5 = y \). Substituting this into the equation, we get:

\[
z

Solve the equation: \sqrt[5]{\log_3(x)} + \log_3(\sqrt{x}) = 12

This page provides a step-by-step solution to the logarithmic equation \sqrt[5]{\log_3(x)} + \log_3(\sqrt{x}) = 12. The solution involves logarithmic identities, exponentiation, and algebraic substitution, suitable for students in Grades 10-12.

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