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The most commonly used bases in logarithms are 10 and e (natural logarithm). Logarithms play a significant role in scientific and financial calculations, with base 10 and base e being widely used. When faced with finding the value of a logarithm with a different base, the change of base formula can be applied. This formula states that log of a to the base B can be expressed as log of a to the base x over log of B to the base x, where the bases in the numerator and denominator must be the same. It allows for calculations of logarithms with different bases using a calculator.