Sure, let's simplify the given expression [tex]\(3 \log_c(y) + 3 (\log_c(z) - 2 \log_c(x))\)[/tex] step-by-step into a single logarithm.
### Step 1: Distribute the coefficient 3 throughout the expression
We begin by distributing the 3 to both terms within the parentheses:
[tex]\[ 3 \log_c(y) + 3(\log_c(z) - 2 \log_c(x)) = 3 \log_c(y) + 3 \log_c(z) - 6 \log_c(x) \][/tex]
### Step 2: Apply the power rule of logarithms
Next, we use the power rule of logarithms, which states that [tex]\( a \log_b(x) = \log_b(x^a) \)[/tex]. We can apply this rule to each term:
[tex]\[ 3 \log_c(y) = \log_c(y^3) \][/tex]
[tex]\[ 3 \log_c(z) = \log_c(z^3) \][/tex]
[tex]\[ -6 \log_c(x) = \log_c(x^{-6}) = \log_c\left(\frac{1}{x^6}\right) \][/tex]
### Step 3: Combine the logarithms
Using the properties of logarithms, we know that [tex]\( \log_b(m) + \log_b(n) = \log_b(m \cdot n) \)[/tex] and [tex]\( \log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right) \)[/tex]. Apply these to combine the terms:
First, combine the positive terms:
[tex]\[ \log_c(y^3) + \log_c(z^3) = \log_c(y^3 \cdot z^3) = \log_c((yz)^3) \][/tex]
Then, incorporate the negative term:
[tex]\[ \log_c((yz)^3) - \log_c(x^6) = \log_c\left(\frac{(yz)^3}{x^6}\right) \][/tex]
### Final Expression
Thus, the given expression [tex]\(3 \log_c(y) + 3 (\log_c(z) - 2 \log_c(x))\)[/tex] simplifies to:
[tex]\[ \log_c\left(\frac{(yz)^3}{x^6}\right) \][/tex]
So the final expression as a single logarithm is:
[tex]\[ \log_c\left(\frac{y^3 z^3}{x^6}\right) \][/tex]
