To find which expressions are equivalent to the given expression [tex]\( 5 \log_{10} x + \log_{10} 20 - \log_{10} 10 \)[/tex], we can use properties of logarithms to simplify and compare.
### Step-by-Step Solution:
1. Given Expression:
[tex]\( 5 \log_{10} x + \log_{10} 20 - \log_{10} 10 \)[/tex]
2. Combine the first two terms using properties of logarithms:
We know that [tex]\( a \log_{10} b = \log_{10} (b^a) \)[/tex]. So:
[tex]\( 5 \log_{10} x = \log_{10} (x^5) \)[/tex]
This transforms the expression to:
[tex]\( \log_{10} (x^5) + \log_{10} 20 - \log_{10} 10 \)[/tex]
3. Combine the sum of logarithms:
Using the property [tex]\( \log_{10} a + \log_{10} b = \log_{10} (ab) \)[/tex], we combine the first two logarithms:
[tex]\( \log_{10} (x^5) + \log_{10} 20 = \log_{10} (x^5 \cdot 20) \)[/tex]
So the expression becomes:
[tex]\( \log_{10} (x^5 \cdot 20) - \log_{10} 10 \)[/tex]
4. Combine the difference of logarithms:
Using the property [tex]\( \log_{10} a - \log_{10} b = \log_{10} \left(\frac{a}{b}\right) \)[/tex], we combine:
[tex]\( \log_{10} (x^5 \cdot 20) - \log_{10} 10 = \log_{10} \left(\frac{x^5 \cdot 20}{10}\right) \)[/tex]
Simplifying inside the logarithm:
[tex]\(\frac{x^5 \cdot 20}{10} = 2 x^5\)[/tex]
Therefore, the expression simplifies to:
[tex]\( \log_{10} (2 x^5) \)[/tex]
### Comparisons:
Now, let's compare with the given options.
1. [tex]\(\log_{10}(2 x)^5\)[/tex]
- This is not equivalent because it implies [tex]\(\log_{10} \left((2x)^5\right) = 5 \log_{10} (2x)\)[/tex], which simplifies differently.
2. [tex]\(\log_{10} \left(20 x^5\right)-1\)[/tex]
- Simplifying this: [tex]\(\log_{10} \left(20 x^5\right) - \log_{10}(10) = \log_{10} \left(\frac{20 x^5}{10}\right) = \log_{10} \left(2 x^5\right)\)[/tex]. This matches our simplified expression.
3. [tex]\(\log_{10} (10 x)\)[/tex]
- This is not equivalent.
4. [tex]\(\log_{10} \left(2 x^5\right)\)[/tex]
- This exactly matches our simplified expression.
5. [tex]\(\log_{10} (100 x) + 1\)[/tex]
- This can be written as [tex]\(\log_{10} (100 x) + \log_{10} (10) = \log_{10} (1000 x)\)[/tex], which does not match our simplified expression.
### Correct Answers:
The expressions equivalent to the given expression are:
- [tex]\(\log_{10}\left(2 x^5\right)\)[/tex]
- [tex]\(\log_{10} \left(20 x^5\right)-1\)[/tex]
Thus, the correct answers are:
[tex]\(\boxed{4}\)[/tex] and [tex]\(\boxed{2}\)[/tex].
