What does it mean for a hormone to be "active" In my textbook it is stated that the T3 hormone produced at the thyroid gland is 3-5 times more biologically active than T4. How is being active defined and how can this effect be measured?

Activity of a hormone represents the efficacy of the hormone in causing the intended effect.

In case of T3 vs T4, though T4 is the majority fraction, T3 is more active, meaning it is much more efficacious in effecting the results.

The concentration of T4 is 30 - 50 times higher than T31. Though T3 is produced by thyroid it is in minority. The majority of T3 is produced by peripheral conversion of T4 into T3.

Thyroid hormones are transported in the blood vessels by albumin, thyroxine binding prealbumin (also called transthyretin). T4 binds more tightly to these transport proteins than T3. In other words T3 constitutes the majority of the free fraction (unbound to carrier/transport proteins). Only the free form of the hormone is metabolically active. So T3 is 3 to 5 times more active than T4.

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T3 - triiodothyroxine

T4 - tetraiodothyroxine

1: pg. 312, Essentials of surgery, Sanjay Azad