Question
re write f(x) = log2 (32x^6)
Answers
GPT-4o mini
To rewrite the function \( f(x) = \log_2(32x^6) \), we can use the properties of logarithms. Specifically, we can use the product property of logarithms:
\[
\log_b(mn) = \log_b(m) + \log_b(n)
\]
Applying this property to \( f(x) \):
\[
f(x) = \log_2(32) + \log_2(x^6)
\]
We can also use the power property of logarithms, which states:
\[
\log_b(m^n) = n \cdot \log_b(m)
\]
So we can rewrite \( \log_2(x^6) \) as:
\[
\log_2(x^6) = 6 \cdot \log_2(x)
\]
Now, we need to calculate \( \log_2(32) \). Since \( 32 = 2^5 \), we have:
\[
\log_2(32) = 5
\]
Putting it all together, we get:
\[
f(x) = 5 + 6 \cdot \log_2(x)
\]
Thus, the rewritten function is:
\[
f(x) = 5 + 6 \log_2(x)
\]
\[
\log_b(mn) = \log_b(m) + \log_b(n)
\]
Applying this property to \( f(x) \):
\[
f(x) = \log_2(32) + \log_2(x^6)
\]
We can also use the power property of logarithms, which states:
\[
\log_b(m^n) = n \cdot \log_b(m)
\]
So we can rewrite \( \log_2(x^6) \) as:
\[
\log_2(x^6) = 6 \cdot \log_2(x)
\]
Now, we need to calculate \( \log_2(32) \). Since \( 32 = 2^5 \), we have:
\[
\log_2(32) = 5
\]
Putting it all together, we get:
\[
f(x) = 5 + 6 \cdot \log_2(x)
\]
Thus, the rewritten function is:
\[
f(x) = 5 + 6 \log_2(x)
\]