To determine which expressions are equivalent to [tex]\(\log \left(10^5\right)\)[/tex], let's analyze each option step by step.
Given:
[tex]\[
\log \left(10^5\right)
\][/tex]
### Step 1: Simplify the given expression
We know the logarithmic property that states [tex]\(\log (a^b) = b \cdot \log (a)\)[/tex]. Using this property, we can simplify [tex]\(\log \left(10^5\right)\)[/tex]:
[tex]\[
\log \left(10^5\right) = 5 \cdot \log (10)
\][/tex]
### Step 2: Verify each option
#### Option A: [tex]\(5 \cdot \log 10\)[/tex]
[tex]\[
5 \cdot \log 10
\][/tex]
Since we already simplified [tex]\(\log \left(10^5\right)\)[/tex] to [tex]\(5 \cdot \log 10\)[/tex], this expression is indeed equivalent.
#### Option B: [tex]\(5 \cdot 10\)[/tex]
[tex]\[
5 \cdot 10 = 50
\][/tex]
Clearly, [tex]\(50 \neq \log \left(10^5\right)\)[/tex]. Therefore, this expression is not equivalent.
#### Option C: 1
[tex]\[
\log \left(10^5\right) \text{ simplifies to } 5 \cdot \log 10
\][/tex]
It's obvious that [tex]\(1 \neq 5 \cdot \log 10\)[/tex]. Therefore, this expression is not equivalent.
#### Option D: 5
[tex]\[
\log \left(10^5\right) \text{ simplifies to } 5 \cdot \log 10
\][/tex]
And, [tex]\(5 \neq 5 \cdot \log 10\)[/tex]. Therefore, this expression is not equivalent.
### Conclusion
After evaluating each option, the only expression that is equivalent to [tex]\(\log \left(10^5\right)\)[/tex] is:
A. [tex]\(5 \cdot \log 10\)[/tex]
Hence, the correct answer is:
- A: [tex]\(5 \cdot \log 10\)[/tex]
