Using numerical analysis and the Quotient Rule of Logarithms, how can log7(a4b14)

Question

log
7
⁡
(
𝑎
4
𝑏
14
)
be rewritten?(1 point)
Responses

4log7a+14log7b
4
log
7
⁡
𝑎
+
14
log
7
⁡
𝑏
4 logarithm superscript 7 baseline left parenthesis a right parenthesis plus 14 logarithm superscript 7 baseline left parenthesis b right parenthesis

14log7b−4log7a
14
log
7
⁡
𝑏
−
4
log
7
⁡
𝑎
14 logarithm subscript 7 baseline b minus 4 logarithm subscript 7 baseline a

2log7a7log7b
2
log
7
⁡
𝑎
7
log
7
⁡
𝑏
Start Fraction 2 logarithm subscript 7 baseline a over 7 logarithm subscript 7 baseline b End Fraction

4log7a−14log7b

Answer 1

To rewrite \(\log_7(a^4 b^{14})\) using the Quotient Rule of Logarithms, we'll apply the properties of logarithms.

Firstly, we use the product rule: \[ \log_b(mn) = \log_b(m) + \log_b(n) \] So, we can split \(\log_7(a^4 b^{14})\) as follows: \[ \log_7(a^4 b^{14}) = \log_7(a^4) + \log_7(b^{14}) \]

Next, we apply the power rule of logarithms: \[ \log_b(m^k) = k \log_b(m) \] This means we can rewrite \(\log_7(a^4)\) and \(\log_7(b^{14})\): \[ \log_7(a^4) = 4 \log_7(a) \] \[ \log_7(b^{14}) = 14 \log_7(b) \]

Putting it all together, we have: \[ \log_7(a^4 b^{14}) = 4 \log_7(a) + 14 \log_7(b) \]

Thus, the correct response is: 4log7a + 14log7b.