Sure, let’s rewrite the given expression [tex]\( \log a - 2 \log b + 3 \log c \)[/tex] using the properties of logarithms. Here is a step-by-step solution:
### Step 1: Apply the Power Rule
The Power Rule of logarithms states that [tex]\( k \log(x) = \log(x^k) \)[/tex]. We'll apply this rule to the terms involving [tex]\( \log b \)[/tex] and [tex]\( \log c \)[/tex]:
- [tex]\( 2 \log b \)[/tex] can be rewritten as [tex]\( \log(b^2) \)[/tex].
- [tex]\( 3 \log c \)[/tex] can be rewritten as [tex]\( \log(c^3) \)[/tex].
So the expression becomes:
[tex]\[ \log a - \log(b^2) + \log(c^3) \][/tex]
### Step 2: Apply the Quotient and Product Rules
The Quotient Rule of logarithms states that [tex]\( \log(x) - \log(y) = \log\left(\frac{x}{y}\right) \)[/tex], and the Product Rule states that [tex]\( \log(x) + \log(y) = \log(x \cdot y) \)[/tex].
- First, combine [tex]\( \log a \)[/tex] and [tex]\(- \log(b^2)\)[/tex] using the Quotient Rule:
[tex]\[ \log a - \log(b^2) = \log\left(\frac{a}{b^2}\right) \][/tex]
- Next, combine the result with [tex]\( \log(c^3) \)[/tex] using the Product Rule:
[tex]\[ \log\left(\frac{a}{b^2}\right) + \log(c^3) = \log\left(\frac{a}{b^2} \cdot c^3\right) \][/tex]
### Step 3: Write the Final Expression
Putting all this together, the expression [tex]\( \log a - 2 \log b + 3 \log c \)[/tex] simplifies to:
[tex]\[ \log\left(\frac{a c^3}{b^2}\right) \][/tex]
Thus, the simplified form of [tex]\( \log a - 2 \log b + 3 \log c \)[/tex] is:
[tex]\[ \boxed{\log\left(\frac{a c^3}{b^2}\right)} \][/tex]
