Find the value of `(log sqrt(27) + log 8 + log sqrt(1000))/(log 120)`

Answer

To solve the expression \((\log \sqrt{27} + \log 8 + \log \sqrt{1000}) / (\log 120)\), we can break it down step by step.

### Step 1: Simplify the logarithmic terms in the numerator

Using the property of logarithms that states \(\log a^b = b \cdot \log a\), we can simplify each term in the numerator:

1. \(\log \sqrt{27} = \log(27^{1/2}) = \frac{1}{2} \log 27\)
2. \(\log 8 = \log(2^3) = 3 \log 2\)
3. \(\log \sqrt{1000} = \log(1000^{1/2}) = \frac{1}{2} \log 1000\)

Now we can rewrite the numerator:

\[ \frac{1}{2} \log 27 + 3 \log 2 + \frac{1}{2} \log 1000 \]

### Step 2: Further simplify the logarithmic terms

Next, we can express \(\log 27\) and \(\log 1000\):

- \(\log 27 = \log(3^3) = 3 \log 3\)
- \(\log 1000 = \log(10^3) = 3 \log 10 = 3\) (since \(\log 10 = 1\))

Substituting these values back into the numerator gives us:

\[ \frac{1}{2} (3 \log 3) + 3 \log 2 + \frac{1}{2} (3) \]

This simplifies to:

\[ \frac{3}{2} \log 3 + 3 \log 2 + \frac{3}{2} \]

### Step 3: Combine the terms in the numerator

Now we can factor out \(\frac{3}{2}\):

\[ \frac{3}{2} (\log 3 + 2 \log 2 + 1) \]

### Step 4: Simplify the denominator

Now, we need to simplify the denominator \(\log 120\). We can express \(120\) as \(120 = 2^3 \cdot 3 \cdot 5\):

\[ \log 120 = \log(2^3) + \log 3 + \log 5 = 3 \log 2 + \log 3 + \log 5 \]

### Step 5: Substitute back into the expression

Now we can substitute the numerator and denominator back into the expression:

\[ \frac{\frac{3}{2} (\log 3 + 2 \log 2 + 1)}{3 \log 2 + \log 3 + \log 5} \]

### Step 6: Calculate the final value

Now we can simplify this expression. We can evaluate it numerically if we know the values of \(\log 2\), \(\log 3\), and \(\log 5\).

Assuming:
- \(\log 2 \approx 0.301\)
- \(\log 3 \approx 0.477\)
- \(\log 5 \approx 0.699\)

Substituting these values into the numerator:

\[ \frac{3}{2} (0.477 + 2 \cdot 0.301 + 1) = \frac{3}{2} (0.477 + 0.602 + 1) = \frac{3}{2} (2.079) \approx 3.1185 \]

And for the denominator:

\[ 3 \cdot 0.301 + 0.477 + 0.699 = 0.903 + 0.477 + 0.699 = 2.079 \]

Finally, we can compute the entire expression:

\[ \frac{3.1185}{2.079} \approx 1.5 \]

### Final Answer

The value of the expression \((\log \sqrt{27} + \log 8 + \log \sqrt{1000}) / (\log 120)\) is approximately \(1.5\).

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