Question

Which of the following uses the properties of logarithms to correctly rewrite f(x)=log2 (32x^6)

a. f(x) = log2 6x+5
b. f(x) = 6 log2 x+5
c. f(x) = 30 log2 x
d. f(x) = 6 log2 x+30

Answer 1

To rewrite \( f(x) = \log_2 (32x^6) \) using the properties of logarithms, we can use the following properties:

1. \(\log_b (xy) = \log_b x + \log_b y\)
2. \(\log_b (x^k) = k \log_b x\)

Using these properties, we can decompose \( f(x) \):

\[
f(x) = \log_2 (32x^6) = \log_2 (32) + \log_2 (x^6)
\]

Next, we apply the second property to \( \log_2 (x^6) \):

\[
\log_2 (x^6) = 6 \log_2 (x)
\]

Now, we also need to evaluate \( \log_2 (32) \). Since \( 32 = 2^5 \):

\[
\log_2 (32) = \log_2 (2^5) = 5
\]

Putting it all together:

\[
f(x) = 5 + 6 \log_2 (x)
\]

Now, rewriting this to match the options:

\[
f(x) = 6 \log_2 x + 5
\]

Thus, the correct answer is:

**a.** \( f(x) = 6 \log_2 x + 5 \)