Question

rewrite in [(a/b)^2 × c^3] using the properties of logarithms

Answer 1

To rewrite \((\frac{a}{b})^2 \times c^3\) using the properties of logarithms, we can apply the logarithmic identities.

1. Use the property \(\log(xy) = \log x + \log y\) for multiplication.
2. Use the property \(\log(x^n) = n \log x\) for exponents.

First, consider \(y = (\frac{a}{b})^2 \times c^3\). Then the logarithm of \(y\) can be expressed as:

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

Using the product rule of logarithms:

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

Now, apply the power rule to each term:

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

Next, we can further simplify \(\log\left(\frac{a}{b}\right)\) using the quotient rule:

\[
\log\left(\frac{a}{b}\right) = \log a - \log b
\]

So, substituting this back, we have:

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

Distributing the 2 gives:

\[
\log y = 2 \log a - 2 \log b + 3 \log c
\]

Thus, the final expression using the properties of logarithms is:

\[
\log y = 2 \log a - 2 \log b + 3 \log c
\]