Let's analyze [tex]\(x = \log_{10} 25,025,088\)[/tex] and determine which of the given statements are true.
First, recall the properties of logarithms:
1. If [tex]\(y = \log_{10} a\)[/tex], then [tex]\(10^y = a\)[/tex].
2. The logarithm gives the exponent to which the base (in this case, 10) must be raised to yield the number.
Let’s consider the value of [tex]\(25,025,088\)[/tex]:
1. Comparing against 7:
- [tex]\(10^7 = 10,000,000\)[/tex] which is less than [tex]\(25,025,088\)[/tex].
- Therefore, [tex]\( \log_{10} 25,025,088 > 7\)[/tex].
2. Comparing against 8:
- [tex]\(10^8 = 100,000,000\)[/tex] which is greater than [tex]\(25,025,088\)[/tex].
- Therefore, [tex]\( \log_{10} 25,025,088 < 8\)[/tex].
Now, let's verify which options are true with these insights.
- a. [tex]\(x < 8\)[/tex]: This is true because we established that [tex]\( \log_{10} 25,025,088 < 8\)[/tex].
- b. [tex]\(x = 8\)[/tex]: This is false because [tex]\( \log_{10} 25,025,088\)[/tex] is less than 8.
- c. [tex]\(x > 8\)[/tex]: This is false for the same reason; [tex]\( \log_{10} 25,025,088\)[/tex] is less than 8.
- d. [tex]\(x < 7\)[/tex]: This is false, as [tex]\( \log_{10} 25,025,088\)[/tex] is greater than 7.
- e. [tex]\(x > 7\)[/tex]: This is true because we established that [tex]\( \log_{10} 25,025,088 > 7\)[/tex].
Therefore, the true statements are:
[tex]\[ \boxed{a \text{ and } e} \][/tex]
