What mathematical operation is performed to bring variables down from exponents?

Taking the natural logarithm of a variable is indeed the correct operation to bring variables down from exponents. When you apply the natural logarithm, you leverage the property of logarithms that states ( \log(a^b) = b \cdot \log(a) ). This means that if a variable is in an exponent, the logarithm effectively pulls the exponent down in front, allowing you to work with it more easily.

For example, if you have an equation like ( e^x = y ) and you want to solve for ( x ), you can take the natural logarithm of both sides, resulting in ( \ln(y) = x ). This transition makes it straightforward to isolate ( x ).

Other operations like taking the square root or taking the square would alter the expression rather than assist in bringing down variables from exponents. Multiplying by the exponent does not conform to any established mathematical principle related to logarithms and would not yield the correct manipulation of variables in terms of solving equations involving exponents.