Question
How do you simplify the expression log2(8) + log2(4) using the properties of logarithms?
Answers
GPT-4o mini
To simplify the expression \( \log_2(8) + \log_2(4) \) using the properties of logarithms, we can use the property that states:
\[
\log_b(m) + \log_b(n) = \log_b(m \cdot n)
\]
In this case, we can apply this property:
1. First, we rewrite the expression:
\[
\log_2(8) + \log_2(4) = \log_2(8 \cdot 4)
\]
2. Next, we calculate \( 8 \cdot 4 \):
\[
8 \cdot 4 = 32
\]
3. Now, we substitute back into the logarithm:
\[
\log_2(8) + \log_2(4) = \log_2(32)
\]
4. Finally, we can simplify \( \log_2(32) \). We know that \( 32 = 2^5 \), so:
\[
\log_2(32) = \log_2(2^5) = 5
\]
Thus, the simplified expression is:
\[
\log_2(8) + \log_2(4) = 5
\]
\[
\log_b(m) + \log_b(n) = \log_b(m \cdot n)
\]
In this case, we can apply this property:
1. First, we rewrite the expression:
\[
\log_2(8) + \log_2(4) = \log_2(8 \cdot 4)
\]
2. Next, we calculate \( 8 \cdot 4 \):
\[
8 \cdot 4 = 32
\]
3. Now, we substitute back into the logarithm:
\[
\log_2(8) + \log_2(4) = \log_2(32)
\]
4. Finally, we can simplify \( \log_2(32) \). We know that \( 32 = 2^5 \), so:
\[
\log_2(32) = \log_2(2^5) = 5
\]
Thus, the simplified expression is:
\[
\log_2(8) + \log_2(4) = 5
\]