Question
What is the first step in establishing that log(107⋅5)=7+log5
?
?
Answers
GPT-4o mini
To establish the equation \( \log(107 \cdot 5) = 7 + \log 5 \), we can apply the properties of logarithms.
The key property we can use here is the logarithmic identity:
\[
\log(a \cdot b) = \log a + \log b
\]
We can first rewrite the left-hand side:
1. Identify \( a \) and \( b \) in \( \log(107 \cdot 5) \):
- Here, \( a = 107 \) and \( b = 5 \).
2. Thus, we can write:
\[
\log(107 \cdot 5) = \log 107 + \log 5
\]
Next, we relate \( \log 107 \) to \( 7 \):
3. Since \( 107 = 10^7 \), we have:
\[
\log 107 = \log(10^7) = 7
\]
Putting it all together, we have:
\[
\log(107 \cdot 5) = \log 107 + \log 5 = 7 + \log 5
\]
So, the first step is to recognize that we can use the logarithmic property \( \log(a \cdot b) = \log a + \log b \). This shows how the original equation breaks down into parts we can analyze.
The key property we can use here is the logarithmic identity:
\[
\log(a \cdot b) = \log a + \log b
\]
We can first rewrite the left-hand side:
1. Identify \( a \) and \( b \) in \( \log(107 \cdot 5) \):
- Here, \( a = 107 \) and \( b = 5 \).
2. Thus, we can write:
\[
\log(107 \cdot 5) = \log 107 + \log 5
\]
Next, we relate \( \log 107 \) to \( 7 \):
3. Since \( 107 = 10^7 \), we have:
\[
\log 107 = \log(10^7) = 7
\]
Putting it all together, we have:
\[
\log(107 \cdot 5) = \log 107 + \log 5 = 7 + \log 5
\]
So, the first step is to recognize that we can use the logarithmic property \( \log(a \cdot b) = \log a + \log b \). This shows how the original equation breaks down into parts we can analyze.