--- a/src/HOL/Log.thy Mon Nov 16 12:09:59 2009 +0100
+++ b/src/HOL/Log.thy Mon Nov 16 14:32:12 2009 +0000
@@ -87,6 +87,17 @@
lemma log_ln: "ln x = log (exp(1)) x"
by (simp add: log_def)
+lemma DERIV_log: "x > 0 ==> DERIV (%y. log b y) x :> 1 / (ln b * x)"
+ apply (subst log_def)
+ apply (subgoal_tac "(%y. ln y / ln b) = (%y. (1 / ln b) * ln y)")
+ apply (erule ssubst)
+ apply (subgoal_tac "1 / (ln b * x) = (1 / ln b) * (1 / x)")
+ apply (erule ssubst)
+ apply (rule DERIV_cmult)
+ apply (erule DERIV_ln_divide)
+ apply auto
+done
+
lemma powr_log_cancel [simp]:
"[| 0 < a; a \<noteq> 1; 0 < x |] ==> a powr (log a x) = x"
by (simp add: powr_def log_def)
@@ -152,9 +163,20 @@
apply (rule powr_realpow [THEN sym], simp)
done
-lemma ln_pwr: "0 < x ==> 0 < y ==> ln(x powr y) = y * ln x"
+lemma ln_powr: "0 < x ==> 0 < y ==> ln(x powr y) = y * ln x"
by (unfold powr_def, simp)
+lemma log_powr: "0 < x ==> 0 \<le> y ==> log b (x powr y) = y * log b x"
+ apply (case_tac "y = 0")
+ apply force
+ apply (auto simp add: log_def ln_powr field_simps)
+done
+
+lemma log_nat_power: "0 < x ==> log b (x^n) = real n * log b x"
+ apply (subst powr_realpow [symmetric])
+ apply (auto simp add: log_powr)
+done
+
lemma ln_bound: "1 <= x ==> ln x <= x"
apply (subgoal_tac "ln(1 + (x - 1)) <= x - 1")
apply simp
@@ -207,7 +229,7 @@
apply (rule mult_imp_le_div_pos)
apply (assumption)
apply (subst mult_commute)
- apply (subst ln_pwr [THEN sym])
+ apply (subst ln_powr [THEN sym])
apply auto
apply (rule ln_bound)
apply (erule ge_one_powr_ge_zero)