This is a pretty broad question, but here are some suggestions.
Start with some remediation of the notion of a proportionality. In the K-12 curriculum, most students only see direct proportionalities, not proportionalities of the form $y \propto x^n$ with $n\ne 1$. Do an example like a comparison of the number of jelly beans that can fit in one jar compared to another, when the second jar has the same shape but double the linear dimensions. If they don't understand this sort of thing, then it's basically a crippling issue when they get to logarithms. Convince them to reason about things like $y_1/y_2$ and $x_1/x_2$, using an example where using these ratios makes it easy to solve the problem in your head, while not using ratios makes it a mess. E.g., for the jelly bean problem, show how awful it gets if they insist on explicitly writing down the formula for the volume of a cylinder and then just cranking out the algebra. Point out that ratios are unitless, which is a big win. Point out that the factors like pi are destined to cancel out. This sort of thing is necessary, or else they will never talk or think about ratios.
Give them tasks in which they translate the identities into the simplest and most transparent possible example using arithmetic. For example, $\log(a^b)=b\log a$ can be exemplified using $10^3$ and log base 10.
Connect to STEM examples such as pH, radioactive half-life, or the number of bits needed to represent an address on a computer with $n$ bytes of memory. Another computer-sciency example is the number of steps needed in order to look up a word in the dictionary, if you follow an algorithm where you repeatedly bisect.
Do the technique where you extract an unknown exponent from a log-log plot of data points. This is most transparent if you start with artificial data and use log base 10, e.g., the data points are (1,5), (10,500), (100,50000). After showing a simple example like this, show an interesting real one. For example, people have measured the amount of effort required for a team of computer programmers to write n lines of code.
Print out a simplified slide rule and hand out scissors, have them use it to do multiplication.
Pose this question to them. Hey, suppose your friend says, "I learned how to manipulate logs in high school, but I never understood what it was about." Explain to your friend in 10 words or less. My own answer would be something like "Logs turn multiplication into addition."