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Logarithms are used in the royal navy in sonars

They are but it's far wider than that.

The application you mean is the "decibel", the 10X or 20X logarithm of the ratio of two signal intensities or powers - not just in military and commercial sonar, but in acoustics generally, and in electrical engineering such as amplifier design.

The Richter scale and the magnitude scale of star brightness are logarithmic. Common and hyperbolic logarithms crop up in many places - the latter control the expansion or compression of gas in an engine or compressor cylinder, for example.

Exponents also facilitate handling very large & very small numbers by turning them into multiples of plus or minus powers of 10.

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Q: What are some real life application of exponents and logarithms?
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