To solve the problem, we need to simplify the given expression step-by-step and compare it with the provided options:
Given expression:
[tex]\[ 5 \log_{10} x + \log_{10} 20 - \log_{10} 10 \][/tex]
First, recall some logarithmic properties:
1. [tex]\(\log_b(mn) = \log_b(m) + \log_b(n)\)[/tex]
2. [tex]\(\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\)[/tex]
3. [tex]\(a \log_b(m) = \log_b(m^a)\)[/tex]
Now, simplify the given expression:
1. Using the property [tex]\(a \log_{10}(m) = \log_{10}(m^a)\)[/tex]:
[tex]\[ 5 \log_{10} x = \log_{10}(x^5) \][/tex]
2. Now the expression becomes:
[tex]\[ \log_{10}(x^5) + \log_{10}(20) - \log_{10}(10) \][/tex]
3. Using the property [tex]\(\log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right)\)[/tex]:
[tex]\[ \log_{10}(20) - \log_{10}(10) = \log_{10}\left(\frac{20}{10}\right) = \log_{10}(2) \][/tex]
4. The expression now is:
[tex]\[ \log_{10}(x^5) + \log_{10}(2) \][/tex]
5. Using the property [tex]\(\log_b(m) + \log_b(n) = \log_b(mn)\)[/tex]:
[tex]\[ \log_{10}(x^5) + \log_{10}(2) = \log_{10}(2 \cdot x^5) = \log_{10}(2x^5) \][/tex]
We compare this with the provided options:
1. [tex]\(\log_{10}\left(20 x^5\right)-1\)[/tex]:
[tex]\[ \log_{10}\left(20 x^5\right)-1 = \log_{10}\left(20 x^5\right) - \log_{10}(10) \][/tex]
[tex]\[ = \log_{10}\left(\frac{20 x^5}{10}\right) = \log_{10}(2 x^5) \][/tex]
This matches our simplified expression [tex]\(\log_{10}(2 x^5)\)[/tex].
2. [tex]\(\log_{10}(2 x)^5\)[/tex]:
This can be expanded:
[tex]\[ \log_{10}(2 x)^5 = 5 \log_{10}(2 x) \][/tex]
[tex]\[ = 5 (\log_{10}(2) + \log_{10}(x)) \][/tex]
[tex]\[ = 5 \log_{10}(x) + 5 \log_{10}(2) \][/tex]
Since the original problem is:
[tex]\[ 5 \log_{10}(x) + \log_{10}(20) - \log_{10}(10) \][/tex]
We already know:
[tex]\[ \log_{10}(20) - \log_{10}(10) = \log_{10}(2) \][/tex]
Thus:
[tex]\[ 5 \log_{10}(x) + \log_{10}(2) = 5 \log_{10}(x) + 5 \log_{10}(2) \][/tex]
This also matches our equivalent expression.
Therefore, the correct answers are:
[tex]\[
\log_{10}(2 x^5)
\][/tex]
And as expanded interpretations match:
Options:
[tex]\[
\log_{10}\left(20 x^5\right)-1
\][/tex]
[tex]\[
\log_{10}(2 x)^5
\][/tex]
So the correct expressions are:
1. [tex]\(\log_{10}\left(20 x^5\right)-1\)[/tex]
2. [tex]\(\log_{10}(2 x)^5\)[/tex]
