Question

given \\(\log_4 3 \approx 0.792\\) and \\(\log_4 21 \approx 2.196\\), what is \\(\log_4 7\\)?

Explanation:

Step1: Recall logarithm quotient rule

The quotient rule of logarithms states that $\log_b \frac{m}{n} = \log_b m - \log_b n$. We know that $21 = 3\times7$, so we can rewrite $\log_4 21$ as $\log_4 (3\times7)$. But we need $\log_4 7$, so we can also use the fact that $7=\frac{21}{3}$, so $\log_4 7=\log_4 \frac{21}{3}$.

Step2: Apply the quotient rule

Using the quotient rule $\log_b \frac{m}{n} = \log_b m - \log_b n$, with $b = 4$, $m = 21$, and $n = 3$, we get:
$\log_4 7=\log_4 21 - \log_4 3$

Step3: Substitute the given values

We are given that $\log_4 3\approx0.792$ and $\log_4 21\approx2.196$. Substituting these values into the equation:
$\log_4 7\approx2.196 - 0.792$

Step4: Calculate the result

$2.196 - 0.792 = 1.404$

Answer:

1.404