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Find the values of $x$ that satisfy the following inequality: $4^x\le5^3$

Given the following inequality:
$4^x\le5^3$
I am not really sure how to solve this. I think that it would be a combination of logarithms and exponents, I am just a little unsure of the wording.
Thanks in advance for any help.

A:

Since $4^x\le 5^3$ is equivalent to $e^{x\ln 4}\le e^{3\ln 5}$, you want to find the values of $x$ such that $e^{x\ln 4}\le e^{3\ln 5}$, or $e^x\le e^3\cdot 5^3$.
You can do this by using the Lambert W-function $W(z)$ (that is defined implicitly for $z>0$ by $ze^z=z$ and $z=W(z)\ln z$ for $z\gt 0$).
We have $x\ln 4\lt 3\ln 5$, or $x\ln 4\lt 0.792$. Now we want $x\lt -1.443$.
(You can find the precise values by doing a numerical root find.)

The Collet-Palette system for ceramic head fabrication: a study in the monkey.
A system for the fabrication of full-coverage ceramic dental restorations using a friction-fit and glaze-bonded metal-ceramic unit is described. The tolerance method, used in the fabrication of metal-ceramic frameworks, is adapted for fabrication of ceramic prosthetic materials. Previous studies have failed to demonstrate the necessary strength and reliability of ceramic restorations with this method. This investigation was designed to determine the limitations of this technique and the feasibility of fabricating ceramic restorations of sufficient strength for use in a primate model. Five adult male Macaca fascicularis monkeys were used in this study. A series of fixed metal-ceramic frameworks were fabricated. Each framework was bisected, and each half

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