Question

SELECT ALL THAT APPLY!! Which of the following would be equivalent to log332−log32 ? (1 point) Responses log332−log32 the log base 3 of 32 minus the log base 3 of 2 2log34 2 log sub 3 4 log38+log38 the log base 3 of 8 plus the log base 3 of 8 log32+log38

Answer 1

To simplify the expression \( \log_3 32 - \log_3 2 \), we can use the logarithmic property that states \( \log_b m - \log_b n = \log_b \left(\frac{m}{n}\right) \). Thus:

\[
\log_3 32 - \log_3 2 = \log_3 \left(\frac{32}{2}\right) = \log_3 16
\]

Now let's analyze the other options:

1. **log332−log32**: This is the original expression, so it is equivalent.
2. **2log34**: Using the property that \( \log_b (m^n) = n \cdot \log_b m \), we have:
\[
2 \log_3 4 = \log_3 (4^2) = \log_3 16
\]
So this is equivalent too.
3. **log38 + log38**: This can be simplified as:
\[
\log_3 8 + \log_3 8 = 2 \log_3 8
\]
This is not equal to \( \log_3 16 \) since \( 8 \) to the power of 2 is \( 64 \) which is different than \( 16 \). So this is not equivalent.
4. **log32 + log38**: This expression can be simplified using properties of logarithms:
\[
\log_3 2 + \log_3 8 = \log_3 (2 \times 8) = \log_3 16
\]
Thus, this is equivalent too.

Based on the above analysis, the equivalent expressions to \( \log_3 32 - \log_3 2 \) are:

- **log332−log32**
- **2log34**
- **log32 + log38**

So the correct selections are: **log332−log32**, **2log34**, and **log32 + log38**.