We are given the equation:

\[
\log \sqrt{27^m} = 3^{1/3}
\]

Let's solve for \( m \) step by step.

### Step 1: Simplify the logarithmic expression
We can rewrite the left-hand side as:

\[
\log \sqrt{27^m} = \log \left(27^m\right)^{1/2}
\]

Since \(\sqrt{27^m} = \left(27^m\right)^{1/2}\), this becomes:

\[
\log \left(27^{m/2}\right)
\]

### Step 2: Use the logarithmic rule
Using the logarithmic rule \(\log (a^b) = b \log a\), we get:

\[
\log \left(27^{m/2}\right) = \frac{m}{2} \log 27
\]

### Step 3: Set equal to the right-hand side
Now, set this equal to \( 3^{1/3} \):

\[
\frac{m}{2} \log 27 = 3^{1/3}
\]

### Step 4: Simplify \( \log 27 \)
Since \( 27 = 3^3 \), we can use the logarithmic rule \(\log a^b = b \log a\), and write:

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

Thus, the equation becomes:

\[
\frac{m}{2} (3 \log 3) = 3^{1/3}
\]

### Step 5: Solve for \( m \)
Now, divide both sides by \( 3 \log 3 \):

\[
\frac{m}{2} = \frac{3^{1/3}}{3 \log 3}
\]

Multiplying both sides by 2:

\[
m = \frac{2 \cdot 3^{1/3}}{3 \log 3}
\]

This is the solution for \( m \).

Would you like further details or have any questions?

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Here are 5 related questions:
1. How do logarithmic rules simplify expressions like \( \log \left(27^{m/2}\right) \)?
2. How does the property \( \log(a^b) = b \log a \) help in solving logarithmic equations?
3. Can you explain why \( \log 27 = 3 \log 3 \) holds true?
4. How would the result change if the base of the logarithm was different?
5. How do fractional exponents, such as \( 3^{1/3} \), affect logarithmic equations?

**Tip**: Remember that logarithmic and exponential rules are closely related; knowing one helps you solve the other easily!

Learn how to solve the logarithmic equation log√27^m=3^(1/3) to find the value of m. This detailed step-by-step solution explores key mathematical concepts such as logarithms, exponents, and logarithmic properties. Suitable for students in Grades 10-12.

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