Rewrite the following as a single logarithmic expression: 3 log subscript b baseline m minus 2 log subscript b baseline n.

Question

(1 point)
Responses

log subscript b baseline left parenthesis start fraction 3m over 2n right parenthesis end fraction
Image with alt text: log subscript b baseline left parenthesis start fraction 3m over 2n right parenthesis end fraction

log subscript b baseline left parenthesis m superscript 3 baseline times n superscript 2 baseline right parenthesis
Image with alt text: log subscript b baseline left parenthesis m superscript 3 baseline times n superscript 2 baseline right parenthesis

log subscript b baseline left parenthesis start fraction m superscript 3 baseline over n superscript 2 baseline right parenthesis end fraction
Image with alt text: log subscript b baseline left parenthesis start fraction m superscript 3 baseline over n superscript 2 baseline right parenthesis end fraction

log subscript b baseline left parenthesis m superscript 3 baseline minus m superscript 2 baseline right parenthesis
Image with alt text: log subscript b baseline left parenthesis m superscript 3 baseline minus m superscript 2 baseline right parenthesis

Answer 1

To rewrite the expression \(3 \log_b m - 2 \log_b n\) as a single logarithmic expression, we can use the properties of logarithms:

1. The power rule: \(k \log_b x = \log_b (x^k)\)
2. The difference rule: \(\log_b x - \log_b y = \log_b \left(\frac{x}{y}\right)\)

Using these properties:

\(3 \log_b m = \log_b (m^3)\)

\(2 \log_b n = \log_b (n^2)\)

Now, substituting these into the original expression:

\[ 3 \log_b m - 2 \log_b n = \log_b (m^3) - \log_b (n^2) = \log_b \left(\frac{m^3}{n^2}\right) \]

Thus, the expression can be rewritten as:

\[ \log_b \left(\frac{m^3}{n^2}\right) \]

The correct response is:

log subscript b baseline left parenthesis start fraction m superscript 3 baseline over n superscript 2 baseline right parenthesis end fraction