Answer :
To determine which expressions are equivalent to the given expression [tex]\( 5 \log_{10} x + \log_{10} 20 - \log_{10} 10 \)[/tex], we will use properties of logarithms to simplify step by step.
Consider the original expression:
[tex]\[ 5 \log_{10} x + \log_{10} 20 - \log_{10} 10 \][/tex]
### Step-by-Step Simplification:
1. Combine the logarithms with subtraction:
We know that [tex]\( \log a - \log b = \log \left(\frac{a}{b}\right) \)[/tex]. So,
[tex]\[ \log_{10} 20 - \log_{10} 10 = \log_{10} \left(\frac{20}{10}\right) = \log_{10} 2 \][/tex]
2. Simplifying the expression:
Now substitute back into the original expression:
[tex]\[ 5 \log_{10} x + \log_{10} 2 \][/tex]
3. Combine the logarithms with addition:
We know that [tex]\( a \log_{10} b = \log_{10} (b^a) \)[/tex] and [tex]\( \log a + \log b = \log (a \cdot b) \)[/tex].Thus:
[tex]\[ 5 \log_{10} x = \log_{10} (x^5) \][/tex]
So the expression becomes:
[tex]\[ \log_{10} (x^5) + \log_{10} 2 = \log_{10} (2 \cdot x^5) = \log_{10} (2x^5) \][/tex]
### Verification of the given options:
1. [tex]\(\log_{10} (2x^5)\)[/tex]:
[tex]\[\log_{10} (2x^5)\][/tex] is equivalent to the simplified expression.
Correct
2. [tex]\(\log_{10} (20x^5) - 1\)[/tex]:
[tex]\[ \log_{10} (20x^5) - 1 = \log_{10} (20x^5) - \log_{10} 10 = \log_{10} \left(\frac{20x^5}{10}\right) = \log_{10} (2x^5) \][/tex]
This is equivalent to the simplified expression.
Correct
3. [tex]\(\log_{10} (10x)\)[/tex]:
[tex]\(\log_{10} (10x)\)[/tex] is not equivalent to the simplified expression [tex]\(\log_{10} (2x^5)\)[/tex].
Incorrect
4. [tex]\(\log_{10} (2x)^5\)[/tex]:
[tex]\[\log_{10} (2x)^5 = \log_{10} (32x^5)\][/tex] is not equivalent to [tex]\(\log_{10} (2x^5)\)[/tex].
Incorrect
5. [tex]\(\log_{10} (100x) + 1\)[/tex]:
[tex]\[ \log_{10} (100x) + 1 = \log_{10} (100x) + \log_{10} 10 = \log_{10} (1000x) \][/tex]
This is not equivalent to [tex]\(\log_{10} (2x^5)\)[/tex].
Incorrect
### Conclusion:
The correct equivalent expressions are:
- [tex]\(\log_{10} (2x^5)\)[/tex]
- [tex]\(\log_{10} (20x^5) - 1\)[/tex]
Consider the original expression:
[tex]\[ 5 \log_{10} x + \log_{10} 20 - \log_{10} 10 \][/tex]
### Step-by-Step Simplification:
1. Combine the logarithms with subtraction:
We know that [tex]\( \log a - \log b = \log \left(\frac{a}{b}\right) \)[/tex]. So,
[tex]\[ \log_{10} 20 - \log_{10} 10 = \log_{10} \left(\frac{20}{10}\right) = \log_{10} 2 \][/tex]
2. Simplifying the expression:
Now substitute back into the original expression:
[tex]\[ 5 \log_{10} x + \log_{10} 2 \][/tex]
3. Combine the logarithms with addition:
We know that [tex]\( a \log_{10} b = \log_{10} (b^a) \)[/tex] and [tex]\( \log a + \log b = \log (a \cdot b) \)[/tex].Thus:
[tex]\[ 5 \log_{10} x = \log_{10} (x^5) \][/tex]
So the expression becomes:
[tex]\[ \log_{10} (x^5) + \log_{10} 2 = \log_{10} (2 \cdot x^5) = \log_{10} (2x^5) \][/tex]
### Verification of the given options:
1. [tex]\(\log_{10} (2x^5)\)[/tex]:
[tex]\[\log_{10} (2x^5)\][/tex] is equivalent to the simplified expression.
Correct
2. [tex]\(\log_{10} (20x^5) - 1\)[/tex]:
[tex]\[ \log_{10} (20x^5) - 1 = \log_{10} (20x^5) - \log_{10} 10 = \log_{10} \left(\frac{20x^5}{10}\right) = \log_{10} (2x^5) \][/tex]
This is equivalent to the simplified expression.
Correct
3. [tex]\(\log_{10} (10x)\)[/tex]:
[tex]\(\log_{10} (10x)\)[/tex] is not equivalent to the simplified expression [tex]\(\log_{10} (2x^5)\)[/tex].
Incorrect
4. [tex]\(\log_{10} (2x)^5\)[/tex]:
[tex]\[\log_{10} (2x)^5 = \log_{10} (32x^5)\][/tex] is not equivalent to [tex]\(\log_{10} (2x^5)\)[/tex].
Incorrect
5. [tex]\(\log_{10} (100x) + 1\)[/tex]:
[tex]\[ \log_{10} (100x) + 1 = \log_{10} (100x) + \log_{10} 10 = \log_{10} (1000x) \][/tex]
This is not equivalent to [tex]\(\log_{10} (2x^5)\)[/tex].
Incorrect
### Conclusion:
The correct equivalent expressions are:
- [tex]\(\log_{10} (2x^5)\)[/tex]
- [tex]\(\log_{10} (20x^5) - 1\)[/tex]
