Question

rewrite in [(a/b)^2 × c^3] using the properties of logarithms
a) 2 x 1n a -2 x 1nb-3 x 1nc
b) (2 x 1n a -2 x 1nb) x 3 x 1nc
c) 2 x 1na-2 x 1nb+3x1nc
d) 2xna-1nb+3 x 1nc

Answer 1

To rewrite \[(\frac{a}{b})^2 \times c^3\] using properties of logarithms, first, let's break it down into logarithmic terms:

1. Using the property of logarithms that states \(\log(xy) = \log(x) + \log(y)\),
2. and the property that states \(\log(x^n) = n\log(x)\).

From the expression \[(\frac{a}{b})^2 \times c^3\], we can take the logarithm:

\[
\ln\left((\frac{a}{b})^2 \times c^3\right) = \ln\left((\frac{a}{b})^2\right) + \ln(c^3)
\]

Now, applying the properties:

\[
= 2 \ln\left(\frac{a}{b}\right) + 3 \ln(c)
\]

Now using the property \(\ln(\frac{x}{y}) = \ln(x) - \ln(y)\):

\[
= 2 (\ln(a) - \ln(b)) + 3 \ln(c)
\]

Distributing the \(2\):

\[
= 2 \ln(a) - 2 \ln(b) + 3 \ln(c)
\]

Therefore, we can summarize:

\[
\ln\left((\frac{a}{b})^2 \times c^3\right) = 2 \ln(a) - 2 \ln(b) + 3 \ln(c)
\]

Now let's match it with the answer choices provided:

a) \(2 \ln a - 2 \ln b - 3 \ln c\)

b) \((2 \ln a - 2 \ln b) \times 3 \ln c\)

c) \(2 \ln a - 2 \ln b + 3 \ln c\)

d) \(2 \ln a - 1 \ln b + 3 \ln c\)

The correct transformation we derived is:

**c)** \(2 \ln a - 2 \ln b + 3 \ln c\)