Question

3 write the expression as a single logarithm.
4 log x - 6 log (x + 2)
a) log x (x + 2)^24
b) 24 log (x/(x + 2))
c) none of these
d) log x^4 (x + 2)^6

Explanation:

Step1: Apply Power Rule of Logarithms

The power rule states that \( n\log a=\log a^n \). For \( 4\log x \), we get \( \log x^4 \), and for \( 6\log(x + 2) \), we get \( \log(x + 2)^6 \). So the expression becomes \( \log x^4-\log(x + 2)^6 \).

Step2: Apply Quotient Rule of Logarithms

The quotient rule states that \( \log a-\log b=\log\frac{a}{b} \). Applying this to \( \log x^4-\log(x + 2)^6 \), we get \( \log\frac{x^4}{(x + 2)^6} \). Now we check the options:

• Option A: \( \log x(x + 2)^{24} \) is incorrect as it doesn't match our result.
• Option B: \( 24\log\frac{x}{x + 2} \) is incorrect as we don't have a factor of 24 from the original coefficients (4 and 6, not 24).
• Option D: \( \log x^4(x + 2)^6 \) is incorrect as it's a product, not a quotient.

So the correct option is C.

Answer:

C. none of these