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Home Exponential and Logartihmic Functions Exponential Functions Rules for Exponents
 
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Rules for Exponents

The following rules for exponents should be familiar to the student when the exponents are rational, except for inequality (vii). They can be proved for real exponents by forming hyperrational exponents and taking standard parts.

RULES FOR EXPONENTS

Let a, b be positive real numbers.

(i)    1x = 1, a0 = 1.

(ii)    ax+y = ax ay, ax-y = ax/ay.

(iii)    axy = (ax)y.

(iv)    axbx = (ab)x (ax/bx) = (a/b)x.

INEQUALITIES FOR EXPONENTS

Let a, b be positive real numbers.

(v) If a < b and x > 0, then ax < bx.

(vi) If 1 < a and x < y, then ax < ay.

(vii) If x ≥ 1, then (a + 1)x ≥ ax + 1.

PROOF

(vii) Since this inequality is probably new to the student, we give a proof for the case where x is a rational number x = q.

Replace a by the variable t. Let

y = (t + 1)q - tq - 1.

We must show that y ≥ 0. When t = 0, y = 0. For t ≥ 0 and g ≥ 1, we have

08_exp-log_functions-5.gif

Thus dy/dt ≥ 0, so y is increasing and y ≥ 0.
Home Exponential and Logartihmic Functions Exponential Functions Rules for Exponents

Last Update: 2006-11-05