Answer :

To compare [tex]\log_5 7[/tex] and [tex]\log_5 \left(\frac{5}{7}\right)[/tex], let's break down the expressions using properties of logarithms.

  1. Understanding the expressions:

    • [tex]\log_5 7[/tex]: This is the logarithm of 7 to the base 5, which asks, "To what power must 5 be raised to result in 7?"
    • [tex]\log_5 \left(\frac{5}{7}\right)[/tex]: This is the logarithm of [tex]\frac{5}{7}[/tex] to the base 5, which asks, "To what power must 5 be raised to result in [tex]\frac{5}{7}[/tex]?"

  2. Using the property of logarithms:

    • We have the property that [tex]\log_b\left(\frac{m}{n}\right) = \log_b m - \log_b n[/tex]. Therefore, [tex]\log_5 \left(\frac{5}{7}\right) = \log_5 5 - \log_5 7[/tex].
    • We know that [tex]\log_5 5 = 1[/tex], because any logarithm of a number to its own base equals 1.

  3. Simplifying the second logarithm:

    • Thus, [tex]\log_5 \left(\frac{5}{7}\right) = 1 - \log_5 7[/tex].

  4. Comparing the two expressions:

    • We need to compare [tex]\log_5 7[/tex] and [tex]1 - \log_5 7[/tex].
    • Clearly, [tex]\log_5 7[/tex] is positive since 7 is greater than 5, making [tex]1 - \log_5 7[/tex] less than 1.
    • Therefore, [tex]\log_5 \left(\frac{5}{7}\right) = 1 - \log_5 7 < \log_5 7[/tex].

In conclusion, [tex]\log_5 \left(\frac{5}{7}\right)[/tex] is less than [tex]\log_5 7[/tex].