Question

Which of the following uses the properties of logarithms to correctly rewrite g(x)=log_2 (8x^-5)
g(x) = -5 log_2 x+3
g(x) = -15_2 x
g(x)= 5 log_2 x+3
g(x)= -5 log_2 x-15

Answer 1

To rewrite the function \( g(x) = \log_2(8x^{-5}) \) using the properties of logarithms, we can break it down as follows:

1. **Use the product property of logarithms**:
\[
\log_b(mn) = \log_b(m) + \log_b(n)
\]
So we can rewrite:
\[
g(x) = \log_2(8) + \log_2(x^{-5})
\]

2. **Next, use the power property of logarithms**:
\[
\log_b(m^n) = n \log_b(m)
\]
Therefore:
\[
\log_2(x^{-5}) = -5 \log_2(x)
\]

3. **Now calculate \( \log_2(8) \)**:
Since \( 8 = 2^3 \):
\[
\log_2(8) = 3
\]

4. **Combine the results**:
Substitute back into the equation:
\[
g(x) = \log_2(8) + \log_2(x^{-5}) = 3 - 5 \log_2(x)
\]
This can be written as:
\[
g(x) = -5 \log_2(x) + 3
\]

Thus, the correct rewritten version of \( g(x) \) is:
\[
g(x) = -5 \log_2 x + 3
\]

Therefore, the correct answer is:
\[
g(x) = -5 \log_2 x + 3
\]