To solve the given problem, let's simplify the given expression step-by-step and then compare it with each of the provided options:
Given expression:
[tex]\[ 5 \log_{10} x + \log_{10} 20 - \log_{10} 10 \][/tex]
Let's break it down:
1. Simplify [tex]\(\log_{10} 10\)[/tex]:
[tex]\[ \log_{10} 10 = 1 \][/tex]
Therefore, the expression becomes:
[tex]\[ 5 \log_{10} x + \log_{10} 20 - 1 \][/tex]
2. Use properties of logarithms:
- The product rule: [tex]\(\log_{10} a + \log_{10} b = \log_{10} (a \cdot b)\)[/tex]
- The quotient rule: [tex]\(\log_{10} a - \log_{10} b = \log_{10} \left( \frac{a}{b} \right)\)[/tex]
- The power rule: [tex]\( n \log_{10} a = \log_{10} a^n \)[/tex]
3. Apply the product rule to combine the logarithms:
[tex]\[ \log_{10} 20 \][/tex]
Using the power rule on [tex]\(5 \log_{10} x\)[/tex]:
[tex]\[ 5 \log_{10} x = \log_{10} x^5 \][/tex]
Combine the logs:
[tex]\[ \log_{10} x^5 + \log_{10} 20 = \log_{10} (20 x^5) \][/tex]
Now, incorporate the subtraction of [tex]\(\log_{10} 10\)[/tex]:
[tex]\[ \log_{10} (20 x^5) - 1 \][/tex]
4. Apply the quotient rule:
[tex]\[ \log_{10} (20 x^5) - \log_{10} 10 = \log_{10} \left( \frac{20 x^5}{10} \right) \][/tex]
[tex]\[ \log_{10} \left( 2 x^5 \right) \][/tex]
So the simplified form of the given expression is:
[tex]\[ \log_{10} (2 x^5) \][/tex]
Now, let's compare this with each of the provided options:
1. [tex]\(\log_{10} (100 x) + 1\)[/tex]:
This does not match our answer, as [tex]\(100x \neq 2x^5\)[/tex].
2. [tex]\(\log_{10} (2 x^5)\)[/tex]:
This expression exactly matches the simplified form. Therefore, it is correct.
3. [tex]\(\log_{10} (20 x^5) - 1\)[/tex]:
Simplifying [tex]\(\log_{10} (20 x^5) - 1\)[/tex] gives us:
[tex]\[ \log_{10} (20 x^5) - \log_{10} 10 = \log_{10} \left( \frac{20 x^5}{10} \right) = \log_{10} (2 x^5) \][/tex]
Therefore, this option also matches our simplified expression.
4. [tex]\(\log_{10} (10 x)\)[/tex]:
This does not match our answer, as [tex]\(10x \neq 2x^5\)[/tex].
5. [tex]\(\log_{10} (2 x)^5\)[/tex]:
Using the rules of logarithms:
[tex]\[ \log_{10} (2 x)^5 = 5 \log_{10} (2 x) \neq \log_{10} (2 x^5) \][/tex]
This does not match our simplified expression.
Therefore, the correct answers are:
[tex]\[ \boxed{2} \][/tex]
