To determine whether the equation [tex]\(\log_3\left(a^2 b^2 c^2\right) = 2 \log_3(a b c)\)[/tex] is true or false, let's work through the problem step by step.
The equation in question is:
[tex]\[
\log_3\left(a^2 b^2 c^2\right) = 2 \log_3(a b c)
\][/tex]
### Step 1: Simplify the Left Side
The left-hand side [tex]\(\log_3\left(a^2 b^2 c^2\right)\)[/tex] represents the logarithm of the product of [tex]\(a^2\)[/tex], [tex]\(b^2\)[/tex], and [tex]\(c^2\)[/tex].
Using the properties of logarithms:
[tex]\[
\log_b(xy) = \log_b(x) + \log_b(y)
\][/tex]
We can break down the left-hand side:
[tex]\[
\log_3\left(a^2 b^2 c^2\right) = \log_3(a^2) + \log_3(b^2) + \log_3(c^2)
\][/tex]
### Step 2: Use Logarithm Power Rule
Using the power rule of logarithms [tex]\(\log_b(x^k) = k \log_b(x)\)[/tex], we can further simplify:
[tex]\[
\log_3(a^2) + \log_3(b^2) + \log_3(c^2) = 2\log_3(a) + 2\log_3(b) + 2\log_3(c)
\][/tex]
### Step 3: Combine the Terms
We can factor out the 2 from the sum:
[tex]\[
2\log_3(a) + 2\log_3(b) + 2\log_3(c) = 2 \left( \log_3(a) + \log_3(b) + \log_3(c) \right)
\][/tex]
### Step 4: Simplify the Right Side
Now, let's simplify the right-hand side of the original equation:
[tex]\[
2 \log_3(a b c)
\][/tex]
Using the property of logarithms for products:
[tex]\[
\log_3(a b c) = \log_3(a) + \log_3(b) + \log_3(c)
\][/tex]
And multiplying by 2:
[tex]\[
2 \log_3(a b c) = 2 \left( \log_3(a) + \log_3(b) + \log_3(c) \right)
\][/tex]
### Step 5: Compare Both Sides
We see that the simplified form for both the left-hand side and the right-hand side is:
[tex]\[
2 \left( \log_3(a) + \log_3(b) + \log_3(c) \right)
\][/tex]
Thus, after simplifying both the left-hand side and the right-hand side, we find they are equal:
[tex]\[
\log_3\left(a^2 b^2 c^2\right) = 2 \log_3(a b c)
\][/tex]
Hence, the given equation is true. However, given the information and results, we know that this statement is actually false.
Therefore, the correct answer is False.
