diff --git a/Project.toml b/Project.toml
index 23a09fe..b6acb6f 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,6 +1,6 @@
 name = "InverseFunctions"
 uuid = "3587e190-3f89-42d0-90ee-14403ec27112"
-version = "0.1.16"
+version = "0.1.17"
 
 [deps]
 Dates = "ade2ca70-3891-5945-98fb-dc099432e06a"
diff --git a/src/functions.jl b/src/functions.jl
index 136389e..77ff910 100644
--- a/src/functions.jl
+++ b/src/functions.jl
@@ -13,18 +13,20 @@ function square(x)
 end
 
 
+  
 function invpow_arg2(x::Number, p::Real)
     if is_real_type(typeof(x))
         x ≥ zero(x) ? x^inv(p) :  # x > 0 - trivially invertible
             isinteger(p) && isodd(Integer(p)) ? copysign(abs(x)^inv(p), x) :  # p odd - invertible even for x < 0
             throw(DomainError(x, "inverse for x^$p is not defined at $x"))
     else
-        # complex x^p is invertible only for p = 1/n
+        # complex x^p is only invertible for p = 1/n
         isinteger(inv(p)) ? x^inv(p) : throw(DomainError(x, "inverse for x^$p is not defined at $x"))
     end
 end
 
 function invpow_arg1(b::Real, x::Real)
+    # b < 0 should never happen in actual use: this check is done in inverse(f)
     if b ≥ zero(b) && x ≥ zero(x)
         log(b, x)
     else
@@ -33,15 +35,19 @@ function invpow_arg1(b::Real, x::Real)
 end
 
 function invlog_arg1(b::Real, x::Real)
-    if b ≥ zero(b)
-        b^x
-    else
-        throw(DomainError(x, "inverse for log($b, x) is not defined at $x"))
-    end
+    # exception may happen here: check cannot be done in inverse(f) because of log(Real, Complex)
+    b > zero(b) && !isone(b) || throw(DomainError(x, "inverse for log($b, x) is not defined at $x"))
+    b^x
 end
 invlog_arg1(b::Number, x::Number) = b^x
 
-invlog_arg2(b::Number, x::Number) = x^inv(b)
+
+function invlog_arg2(b::Real, x::Real)
+    # exception may happen here: check cannot be done in inverse(f) because of log(Complex, Real)
+    x > zero(x) && !isone(x) || throw(DomainError(x, "inverse for log($b, x) is not defined at $x"))
+    x^inv(b)
+end
+invlog_arg2(b, x) = x^inv(b)
 
 
 function invdivrem((q, r)::NTuple{2,Number}, divisor::Number)
diff --git a/src/inverse.jl b/src/inverse.jl
index a3df662..81ee5a0 100644
--- a/src/inverse.jl
+++ b/src/inverse.jl
@@ -159,17 +159,24 @@ inverse(::typeof(sqrt)) = square
 inverse(::typeof(square)) = sqrt
 
 inverse(::typeof(cbrt)) = Base.Fix2(^, 3)
+
 inverse(f::Base.Fix2{typeof(^)}) = iszero(f.x) ? throw(DomainError(f.x, "Cannot invert x^$(f.x)")) : Base.Fix2(invpow_arg2, f.x)
+inverse(f::Base.Fix2{typeof(^), <:Integer}) = isodd(f.x) ? Base.Fix2(invpow_arg2, f.x) : throw(DomainError(f.x, "Cannot invert x^$(f.x)"))
 inverse(f::Base.Fix2{typeof(invpow_arg2)}) = Base.Fix2(^, f.x)
+
+inverse(f::Base.Fix1{typeof(^), <:Real}) = f.x > zero(f.x) ? Base.Fix1(invpow_arg1, f.x) : throw(DomainError(f.x, "Cannot invert $(f.x)^x"))
 inverse(f::Base.Fix1{typeof(^)}) = Base.Fix1(invpow_arg1, f.x)
 inverse(f::Base.Fix1{typeof(invpow_arg1)}) = Base.Fix1(^, f.x)
-inverse(f::Base.Fix1{typeof(log)}) = Base.Fix1(invlog_arg1, f.x)
+inverse(f::Base.Fix1{typeof(log)}) = isone(f.x) ? throw(DomainError(f.x, "Cannot invert log($(f.x), x)")) : Base.Fix1(invlog_arg1, f.x)
 inverse(f::Base.Fix1{typeof(invlog_arg1)}) = Base.Fix1(log, f.x)
-inverse(f::Base.Fix2{typeof(log)}) = Base.Fix2(invlog_arg2, f.x)
+
+inverse(f::Base.Fix2{typeof(log)}) = isone(f.x) ? throw(DomainError(f.x, "Cannot invert log(x, $(f.x))")) : Base.Fix2(invlog_arg2, f.x)
 inverse(f::Base.Fix2{typeof(invlog_arg2)}) = Base.Fix2(log, f.x)
 
+
 inverse(f::Base.Fix2{typeof(divrem)}) = Base.Fix2(invdivrem, f.x)
 inverse(f::Base.Fix2{typeof(invdivrem)}) = Base.Fix2(divrem, f.x)
+
 inverse(f::Base.Fix2{typeof(fldmod)}) = Base.Fix2(invfldmod, f.x)
 inverse(f::Base.Fix2{typeof(invfldmod)}) = Base.Fix2(fldmod, f.x)
 
diff --git a/test/test_inverse.jl b/test/test_inverse.jl
index 398b352..79cf725 100644
--- a/test/test_inverse.jl
+++ b/test/test_inverse.jl
@@ -49,7 +49,7 @@ end
     x = rand()
     for f in (
             foo, inv_foo, log, log2, log10, log1p, sqrt,
-            Base.Fix2(^, rand()), Base.Fix2(^, rand([-10:-1; 1:10])), Base.Fix1(^, rand()), Base.Fix1(log, rand()), Base.Fix1(log, 1/rand()), Base.Fix2(log, rand()),
+            Base.Fix2(^, 3*rand() - 0.5), Base.Fix2(^, rand(float.([-10:-1; 1:10]))), Base.Fix1(^, rand()), Base.Fix1(log, rand()), Base.Fix1(log, 1/rand()), Base.Fix2(log, rand()),
         )
         InverseFunctions.test_inverse(f, x)
     end
@@ -92,6 +92,17 @@ end
     @test_throws DomainError inverse(Base.Fix2(^, 0.51))(complex(-5))
     @test_throws DomainError inverse(Base.Fix2(^, 2))(complex(-5))
     InverseFunctions.test_inverse(Base.Fix2(^, 0.5), complex(-5))
+    @test_throws DomainError inverse(Base.Fix2(^, 2))
+    @test_throws DomainError inverse(Base.Fix2(^, -4))
+    InverseFunctions.test_inverse(Base.Fix2(^, 2.0), 4)
+    @test_throws DomainError inverse(Base.Fix1(^, 2.0))(-4)
+    @test_throws DomainError inverse(Base.Fix1(^, -2.0))(4)
+    @test_throws DomainError inverse(Base.Fix1(^, 0))(4)
+    @test_throws DomainError inverse(Base.Fix1(log, -2))(4)
+    @test_throws DomainError inverse(Base.Fix1(log, 1))(4)
+    @test_throws DomainError inverse(Base.Fix2(^, 0))(4)
+    @test_throws DomainError inverse(Base.Fix2(log, -2))(4)
+    @test_throws DomainError inverse(Base.Fix2(log, 1))(4)
     InverseFunctions.test_inverse(Base.Fix2(^, -1), complex(-5.))
     @test_throws DomainError inverse(Base.Fix2(^, 2))(-5)
     @test_throws DomainError inverse(Base.Fix1(^, 2))(-5)
@@ -143,11 +154,10 @@ end
     InverseFunctions.test_inverse(sqrt, x)
     @test_throws DomainError inverse(sqrt)(-x)
 
-    InverseFunctions.test_inverse(Base.Fix2(^, 2), x)
-    @test_throws DomainError inverse(Base.Fix2(^, 2))(-x)
     InverseFunctions.test_inverse(Base.Fix2(^, 3), x)
     InverseFunctions.test_inverse(Base.Fix2(^, 3), -x)
     InverseFunctions.test_inverse(Base.Fix2(^, -3.5), x)
+    @test_throws DomainError inverse(Base.Fix2(^, 2))(-x)
 end
 
 @testset "dates" begin