We are asked to simplify the expression $3\log 8 - 3\log 6$ and express the result as a single logarithm. We assume the logarithm is base 10.

1. Problem Description

We are asked to simplify the expression $3\log 8 - 3\log 6$ and express the result as a single logarithm. We assume the logarithm is base

1

0.

2. Solution Steps

First, we use the logarithm property $a\log x = \log x^a$ to rewrite the expression:

$3\log 8 - 3\log 6 = \log 8^3 - \log 6^3$

$8^3 = 8 \times 8 \times 8 = 512$

$6^3 = 6 \times 6 \times 6 = 216$

Therefore, we have:

$\log 8^3 - \log 6^3 = \log 512 - \log 216$

Next, we use the logarithm property $\log x - \log y = \log \frac{x}{y}$ to combine the logarithms:

$\log 512 - \log 216 = \log \frac{512}{216}$

We simplify the fraction $\frac{512}{216}$ by dividing both the numerator and the denominator by their greatest common divisor.

Both are divisible by 2:

$\frac{512}{216} = \frac{256}{108}$

Both are divisible by 2:

$\frac{256}{108} = \frac{128}{54}$

Both are divisible by 2:

$\frac{128}{54} = \frac{64}{27}$

So, we have:

$\log \frac{512}{216} = \log \frac{64}{27}$

3. Final Answer

$\log \frac{64}{27}$