Question

Сравнить логарифм 5 по основанию 4 и логарифм 6 по основанию 5

Answer 1

$log_5 6 \ \textless \ log_4 5 \\ \sqrt{ \frac{log_56}{log_45} }= \sqrt{log_56log_54}\ \textless \ \frac{log_56+log_54}{2} \ \textless \ \frac{log_56*4}{2}= \frac{log_520}{2}=log_5 \sqrt{20}= \\ =log_{5}2*5^{ \frac{1}{2} }=log_52+log_55^{ \frac{1}{2}} = \frac{1}{2} +log_52\ \textless \ 1 \\ \frac{log_56}{log_45}\ \textless \ 1 \iff log_56\ \textless \ log_45$

Answer 2

Сравнить $log_56$ и $log_45$ более-менее традиционными и доступными для большей части решателей не удалось, т.к. эти два числа очень близки по значению.
Буду использовать неравенство Коши $\sqrt{ab} \ \textless \ \dfrac{a+b}{2}$, где $a \geq 0, \geq 0.$
У нас как раз $log_56\ \textgreater \ 0\ u\ log_45\ \textgreater \ 0.$
Рассмотрим отношение:
$\sqrt{\dfrac{log_56}{log_45}}=\sqrt{{log_56}*{log_54}} \leq \dfrac{{log_56}+{log_54}}{2} =\dfrac{{log_524}}{2} \ \textless \ \dfrac{{log_525}}{2}= \dfrac{2}{2}=1$
Отсюда, $\sqrt{\dfrac{log_56}{log_45}}\ \textless \ 1\ =\ \textgreater \ \dfrac{log_56}{log_45}\ \textless \ 1=\ \textgreater \ {log_56}\ \textless \ {log_45}$
Ответ: ${log_56}\ \textless \ {log_45}$

Answer 3

Comparing logarithm of 5 base 4 and logarithm of 6 base 5

To compare the logarithm of 5 base 4 and the logarithm of 6 base 5, we can use the property of logarithms that states: log_a(b) = log_c(b) / log_c(a).

Let's calculate the values:

1. Logarithm of 5 base 4: - Using the property mentioned above, we can rewrite it as: log₄(5) = log_c(5) / log_c(4). - Since the base is not specified, we can choose any base for the logarithm. Let's choose base 10 for simplicity. - Using the change of base formula, we can rewrite it as: log₄(5) = log₁₀(5) / log₁₀(4). - Evaluating the logarithms using a calculator, we find that log₄(5) ≈ 0.8614.

2. Logarithm of 6 base 5: - Using the property mentioned above, we can rewrite it as: log₅(6) = log_c(6) / log_c(5). - Again, let's choose base 10 for simplicity. - Using the change of base formula, we can rewrite it as: log₅(6) = log₁₀(6) / log₁₀(5). - Evaluating the logarithms using a calculator, we find that log₅(6) ≈ 0.7782.

Therefore, the logarithm of 5 base 4 is approximately 0.8614, and the logarithm of 6 base 5 is approximately 0.7782.

Conclusion

The logarithm of 5 base 4 is greater than the logarithm of 6 base 5.