Rewrite the expression 2log636−log864

Question

2
log
6
⁡
36
−
log
8
⁡
64
using the properties of logarithms.(1 point)
Responses

0
0

log−2(36264)
log
−
2
⁡
(
36
2
64
)
logarithm subscript negative 2 baseline left parenthesis Start Fraction 36 squared over 64 End Fraction right parenthesis

4
4

2

Answer 1

To rewrite the expression \(2 \log_6 36 - \log_8 64\) using the properties of logarithms, we can apply the following properties:

1. The power rule: \(a \log_b x = \log_b (x^a)\)
2. The change of base rule: \(\log_b x = \frac{\log_k x}{\log_k b}\) (for this problem, we can keep the logarithms in their original bases if applicable)

Let's proceed step by step:

1. Apply the power rule to \(2 \log_6 36\): \[ 2 \log_6 36 = \log_6 (36^2) \]

2. Calculate \(36^2\): \[ 36^2 = 1296 \] Thus, \[ 2 \log_6 36 = \log_6 1296 \]

3. Now consider \(\log_8 64\). Since \(64 = 8^2\), we can use the property of logarithms: \[ \log_8 64 = \log_8 (8^2) = 2 \]

4. Combine the two parts together: We now substitute back into the original expression: \[ 2 \log_6 36 - \log_8 64 = \log_6 1296 - 2 \]

5. Express the subtraction in terms of logarithms: We can rewrite \(2\) as \(\log_6 (6^2)\), since \(2 = \log_6(36)\): \[ 2 = \log_6 36 \] Therefore: \[ \log_6 1296 - \log_6 36 = \log_6 \left(\frac{1296}{36}\right) \]

6. Calculate \(\frac{1296}{36}\): \[ \frac{1296}{36} = 36 \] Hence, \[ \log_6 \left(\frac{1296}{36}\right) = \log_6 36 \]

This gives us the final expression as: \[ \log_6 36 \]

So the rewritten expression can be presented as: \[ \log_6 \left(\frac{36^2}{64}\right) \] or simply, \[ \log_6 36 \]

Thus, the responses:

• Option 1: \(\log_{-2}\left(\frac{36^2}{64}\right)\)
• Option 4: \(4\) (equivalent, if we evaluate \(\log_6 36\))

Based on the responses you provided:

• The expression corresponds to the first interpretation, but note that \(4\) is incorrect in evaluation after steps since that yields finite simple terms.

Please clarify if you need a particular option in the set responses you mentioned!