Comparing logarithms with different bases $\log_3 4$ and $\log_7 10$: which of these two logarithms is greater?

I figured out that both are between $1$ and $2$, then between $1$ and $1.5$. And then $\log_34$ is greater than $1.25$, and $\log_710$ is smaller than $1.25$. However, that method doesn't work for every example, and I wonder if there's a easier way to solve this?
Here is a proof using no numbers larger than $128$. We have $81 &lt; 125 &lt; 128$, i.e. $3^4 &lt; 5^3 &lt; 2^7$, therefore $\log_35 &gt; \tfrac{4}{3}$ and $\log_25 &lt; \tfrac{7}{3}$, therefore $\log_25 &lt; 1 + \log_35$, therefore:
$$
\log_210 &lt; 2 + \log_35 = \log_345 &lt; \log_349 = 2\log_37,
$$
therefore $\log_410 &lt; \log_37$, therefore $\log_34 &gt; \log_710$. $\square$

The last step uses the general proposition that $\log_ab &gt; \log_cd$ if and only if
$\log_bd &lt; \log_ac$, which can be proved by rewriting all the logarithms in terms of logarithms to a single base (e.g. $\log_ab = \ln b/\ln a$, etc.).
We'll  show that $$\log_34&gt;\log_710$$ or
$$4&gt;3^{\log_710},$$ which is true because $$\log_710&lt;1.2$$ and $$4&gt;3^{1.2}.$$