One of them is measuring earthquakes.
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From the Wikipedia article (link below), Presentation of data on a logarithmic scale can be helpful when the data cover a large range of values - for the use of the logarithms of the values rather than the actual values reduces a wide range to a more manageable size.
Natural logarithms use base e (approximately 2.71828), common logarithms use base 10.
Infinitely many. You can represent any distance in real life by 1 unit on your scale or, conversely, you use any distance on your scale drawing to represent a unit of length in real life.
Unless you are an electrical engineer or a math teacher, every number you will ever use in a real world situation will be a real number.
Physicists and Engineers use small scale models like this all the time to convert to real world use. A toy car going down a ramp can simulate how a life size car will respond going down a hill. They can then use that information to build safer roads and vehicles.