The base 10 exponent for a number :

This is the number such that :

The base-2 exponent:

This is the number such that :

The real exponent for an arbitrary-precision number:

The result of RealExponent is given as a machine number whatever the precision of :

The real exponent for an exact number:

The real exponent for an exact numeric quantity:

Real exponent for zeros:

This is -Accuracy[x]:

The same is true for arbitrary-precision zeros:

Also for exact zero:

The real exponent for different bases:

The base can be any number strictly greater than 1:

The base can be an exact numeric quantity:

RealExponent automatically threads over lists:

Determine quickly if a power will overflow:

This predicts that the power can be represented:

A larger power will not work:

Determine quickly the largest power tower that will not overflow for a given number:

For any approximate number x, RealExponent[x] is equal to Precision[x]-Accuracy[x]:

Also true for arbitrary-precision numbers:

If x is an approximate zero, then RealExponent[x] gives -Accuracy[x]:

This is the same as saying that the identity RealExponent[x] is equal to Precision[x]-Accuracy[x]:

Since precision is zero for approximate zeros:

The real exponent of a product is the sum of the real exponents:

The real exponent of a power is the real exponent of the base times the power: