To determine the order of the expressions
- (A) [tex]\(\log_{11}(10)\)[/tex]
- (B) [tex]\(\log_{7}(40)\)[/tex]
- (C) [tex]\(\log_{5}(27)\)[/tex]
- (D) [tex]\(\log_{2}\left(\frac{1}{4}\right)\)[/tex]
from smallest to largest, follow these steps:
1. Convert Each Logarithm to a Common Base:
To compare these logarithms, let's use the change of base formula, which states:
[tex]\[
\log_b(a) = \frac{\log_k(a)}{\log_k(b)}
\][/tex]
where [tex]\(k\)[/tex] is a common base (e.g., the natural logarithm [tex]\(\log\)[/tex] or the base 10 logarithm [tex]\(\log_{10}\)[/tex]). Here, we’ll use [tex]\(\log\)[/tex] (natural logarithm) for simplicity.
For each logarithm expression, apply the change of base formula:
- (A) [tex]\(\log_{11}(10)\)[/tex]:
[tex]\[
\log_{11}(10) = \frac{\log(10)}{\log(11)}
\][/tex]
- (B) [tex]\(\log_{7}(40)\)[/tex]:
[tex]\[
\log_{7}(40) = \frac{\log(40)}{\log(7)}
\][/tex]
- (C) [tex]\(\log_{5}(27)\)[/tex]:
[tex]\[
\log_{5}(27) = \frac{\log(27)}{\log(5)}
\][/tex]
- (D) [tex]\(\log_{2}\left(\frac{1}{4}\right)\)[/tex]:
[tex]\[
\log_{2}\left(\frac{1}{4}\right) = \frac{\log\left(\frac{1}{4}\right)}{\log(2)}
\][/tex]
Since [tex]\(\log\left(\frac{1}{4}\right) = \log(1) - \log(4) = 0 - \log(4) = -\log(4)\)[/tex], it follows that:
[tex]\[
\log_{2}\left(\frac{1}{4}\right) = \frac{-\log(4)}{\log(2)}
\][/tex]
2. Evaluate and Compare Each Expression:
By noting the result of calculations obtained through this evaluation (or by using a numerical approach), we determine the values of each logarithmic expression relative to one another.
3. Sort the Logarithms:
After evaluating, the expressions in increasing order are:
- (D) [tex]\(\log_{2}\left(\frac{1}{4}\right)\)[/tex]
- (A) [tex]\(\log_{11}(10)\)[/tex]
- (B) [tex]\(\log_{7}(40)\)[/tex]
- (C) [tex]\(\log_{5}(27)\)[/tex]
Thus, the expressions in increasing order are:
[tex]\[
D, A, B, C
\][/tex]
