To determine which expressions are equivalent to the given expression [tex]\(5 \log_{10} x + \log_{10} 20 - \log_{10} 10\)[/tex], let's break down and simplify this expression step by step using properties of logarithms.
1. Simplify [tex]\(\log_{10} 20 - \log_{10} 10\)[/tex]:
- Using the property [tex]\(\log_b(a) - \log_b(b) = \log_b\left(\frac{a}{b}\right)\)[/tex], we have:
[tex]\[
\log_{10} 20 - \log_{10} 10 = \log_{10} \left(\frac{20}{10}\right) = \log_{10} 2
\][/tex]
2. Substitute back into the original expression:
- Replace [tex]\(\log_{10} 20 - \log_{10} 10\)[/tex] with [tex]\(\log_{10} 2\)[/tex]:
[tex]\[
5 \log_{10} x + \log_{10} 2
\][/tex]
3. Combine the logarithms:
- Using the property [tex]\(\log_b(a) + \log_b(b) = \log_b(ab)\)[/tex], we combine the logarithms:
[tex]\[
5 \log_{10} x + \log_{10} 2 = \log_{10}(x^5) + \log_{10}(2) = \log_{10}(2 \cdot x^5)
\][/tex]
So, the given expression [tex]\(5 \log_{10} x + \log_{10} 20 - \log_{10} 10\)[/tex] simplifies to [tex]\(\log_{10}(2 \cdot x^5)\)[/tex].
Now, let's analyze the candidate expressions:
1. [tex]\(\log_{10}(2 x^5)\)[/tex]:
- Equivalent. This directly matches our simplified expression.
2. [tex]\(\log_{10}(10 x)\)[/tex]:
- Not equivalent. Using the property [tex]\(\log_b(ab) = \log_b(a) + \log_b(b)\)[/tex], this simplifies to [tex]\(\log_{10} 10 + \log_{10} x = 1 + \log_{10} x\)[/tex], which is not the same as [tex]\(\log_{10}(2 \cdot x^5)\)[/tex].
3. [tex]\(\log_{10}\left(20 x^5\right) - 1\)[/tex]:
- Equivalent. We can process this as:
[tex]\[
\log_{10}(20 x^5) - \log_{10}(10) = \log_{10}\left(\frac{20 x^5}{10}\right) = \log_{10}(2 x^5)
\][/tex]
- Thus, it matches our simplified expression.
4. [tex]\(\log_{10}(2 x)^5\)[/tex]:
- Not equivalent. This represents:
[tex]\[
5 \log_{10}(2 x) = 5(\log_{10} 2 + \log_{10} x) = 5 \log_{10} 2 + 5 \log_{10} x = \log_{10}(2^5) + \log_{10}(x^5) = \log_{10}(32 x^5)
\][/tex]
- This does not simplify to [tex]\(\log_{10}(2 x^5)\)[/tex].
5. [tex]\(\log_{10}(100 x) + 1\)[/tex]:
- Not equivalent. Let's simplify it step-by-step:
[tex]\[
\log_{10}(100 x) + 1 = \log_{10}(10^2 x) + 1 = \log_{10}(100 x) + \log_{10} 10 = \log_{10}(100 x \cdot 10) = \log_{10}(1000 x)
\][/tex]
- This does not simplify to [tex]\(\log_{10}(2 x^5)\)[/tex].
In summary, these are the equivalent expressions:
- [tex]\(\log _{10}(2 x^5)\)[/tex]
- [tex]\(\log _{10}\left(20 x^5\right)-1\)[/tex]
