Which option best describes how to show that the Product Rule of Logarithms works for the base 4 expression log4 16+log4 4

Question

?(1 point)
Responses

Evaluate both log4 16
and log4 4
and show that their sum is equal to the value of log4 64
.
Evaluate both logarithm subscript 4 baseline 16 and logarithm subscript 4 baseline 4 and show that their sum is equal to the value of logarithm subscript 4 baseline 64 .

Evaluate both log4 16
and log4 4
and show that their product is equal to the value of log4 64
.
Evaluate both logarithm subscript 4 baseline 16 and logarithm subscript 4 baseline 4 and show that their product is equal to the value of logarithm subscript 4 baseline 64 .

Evaluate both log4 16
and log4 4
and show that their sum is equal to the value of log4 20
.
Evaluate both logarithm subscript 4 baseline 16 and logarithm subscript 4 baseline 4 and show that their sum is equal to the value of logarithm subscript 4 baseline 20 .

Evaluate both log4 16
and log4 4
and show that their product is equal to the value of log4 20
.

Answer 1

To show that the Product Rule of Logarithms works for the base 4 expression \(\log_4 16 + \log_4 4\), we need to evaluate both logarithms and demonstrate that their sum is equal to the logarithm of the product of their arguments.

Given:

• \(\log_4 16\)
• \(\log_4 4\)

We can evaluate these:

1. \(\log_4 16\): Since \(16 = 4^2\), we have \(\log_4 16 = 2\).
2. \(\log_4 4\): Since \(4 = 4^1\), we find \(\log_4 4 = 1\).

Now, adding these results: \[ \log_4 16 + \log_4 4 = 2 + 1 = 3. \]

Now, we need to check the value of \(\log_4(16 \times 4)\): \[ 16 \times 4 = 64, \] and since \(64 = 4^3\), we have \(\log_4 64 = 3\).

Thus, we find that: \[ \log_4 16 + \log_4 4 = \log_4 64. \]

The best response that describes this conclusion is: Evaluate both \(\log_4 16\) and \(\log_4 4\) and show that their sum is equal to the value of \(\log_4 64\).