diff --git a/Project.toml b/Project.toml
index dd120c59..204fbe47 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,7 +1,7 @@
 name = "MeasureTheory"
 uuid = "eadaa1a4-d27c-401d-8699-e962e1bbc33b"
 authors = ["Chad Scherrer <chad.scherrer@gmail.com> and contributors"]
-version = "0.11.4"
+version = "0.12.0"
 
 [deps]
 Accessors = "7d9f7c33-5ae7-4f3b-8dc6-eff91059b697"
diff --git a/src/MeasureTheory.jl b/src/MeasureTheory.jl
index 9a0280bc..dd6f700a 100644
--- a/src/MeasureTheory.jl
+++ b/src/MeasureTheory.jl
@@ -76,7 +76,7 @@ function logpdf(d::AbstractMeasure, x)
     _logpdf(d, x, logdensity(d,x))
 end
 
-function _logpdf(d::AbstractMeasure, x, acc::Float64)
+function _logpdf(d::AbstractMeasure, x, acc::Real)
     β = basemeasure(d)
     d === β && return acc
 
@@ -104,6 +104,7 @@ include("combinators/weighted.jl")
 include("combinators/product.jl")
 include("combinators/transforms.jl")
 include("combinators/chain.jl")
+# include("combinators/basemeasure.jl")
 
 include("distributions.jl")
 
diff --git a/src/parameterized/exponential.jl b/src/parameterized/exponential.jl
index 3a742d0e..b28c28c5 100644
--- a/src/parameterized/exponential.jl
+++ b/src/parameterized/exponential.jl
@@ -3,9 +3,9 @@
 
 export Exponential
 
-@parameterized Exponential(λ) ≃ Lebesgue(ℝ₊)
+@parameterized Exponential(β) ≃ Lebesgue(ℝ₊)
+
 
-@kwstruct Exponential(λ)
 @kwstruct Exponential()
 
 function logdensity(d::Exponential{()} , x)
@@ -14,21 +14,82 @@ end
 
 Base.rand(rng::AbstractRNG, T::Type, μ::Exponential{()}) = randexp(rng,T)
 
+TV.as(::Exponential) = asℝ₊
+
+
+##########################
+# Scale β
+
+@kwstruct Exponential(β)
+
+function Base.rand(rng::AbstractRNG, T::Type, d::Exponential{(:β,)})
+    randexp(rng, T) * d.β
+end
+
+function logdensity(d::Exponential{(:β,)}, x)
+    z = x / d.β
+    return logdensity(Exponential(), z) - log(d.β)
+end
+
+distproxy(d::Exponential{(:β,)}) = Dists.Exponential(d.β)
+
+asparams(::Type{<:Exponential}, ::Val{:β}) = asℝ₊
+
+##########################
+# Log-Scale logβ
+
+
+@kwstruct Exponential(logβ)
+
+function Base.rand(rng::AbstractRNG, T::Type, d::Exponential{(:logβ,)})
+    randexp(rng, T) * exp(d.logβ)
+end
+
+function logdensity(d::Exponential{(:logβ,)}, x)
+    z = x * exp(-d.logβ)
+    return logdensity(Exponential(), z) - d.logβ
+end
+
+distproxy(d::Exponential{(:logβ,)}) = Dists.Exponential(exp(d.logβ))
+
+asparams(::Type{<:Exponential}, ::Val{:logβ}) = asℝ
+
+
+
 
 ##########################
+# Rate λ
+
+@kwstruct Exponential(λ)
 
 function Base.rand(rng::AbstractRNG, T::Type, d::Exponential{(:λ,)})
     randexp(rng, T) / d.λ
 end
 
-TV.as(::Exponential) = asℝ₊
-
 function logdensity(d::Exponential{(:λ,)}, x)
     z = x * d.λ
     return logdensity(Exponential(), z) + log(d.λ)
 end
 
-distproxy(d::Exponential{(:λ,)}) = Dists.Exponential(d.λ)
+distproxy(d::Exponential{(:λ,)}) = Dists.Exponential(1/d.λ)
 
 asparams(::Type{<:Exponential}, ::Val{:λ}) = asℝ₊
+
+##########################
+# Log-Rate logλ
+
+
+@kwstruct Exponential(logλ)
+
+function Base.rand(rng::AbstractRNG, T::Type, d::Exponential{(:logλ,)})
+    randexp(rng, T) * exp(-d.logλ)
+end
+
+function logdensity(d::Exponential{(:logλ,)}, x)
+    z = x * exp(d.logλ)
+    return logdensity(Exponential(), z) + d.logλ
+end
+
+distproxy(d::Exponential{(:logλ,)}) = Dists.Exponential(exp(-d.logλ))
+
 asparams(::Type{<:Exponential}, ::Val{:logλ}) = asℝ
diff --git a/test/runtests.jl b/test/runtests.jl
index 1390114b..361f045e 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -94,6 +94,20 @@ end
         @test_broken logdensity(Binomial(n,p), CountingMeasure(ℤ[0:n]), x) ≈ binomlogpdf(n,p,x)
     end
 
+    @testset "Exponential" begin
+        r = rand(MersenneTwister(123), Exponential(2))
+        @test r ≈ rand(MersenneTwister(123), Exponential(β=2))
+        @test r ≈ rand(MersenneTwister(123), Exponential(λ=0.5))
+        @test r ≈ rand(MersenneTwister(123), Exponential(logβ=log(2)))
+        @test r ≈ rand(MersenneTwister(123), Exponential(logλ=log(0.5)))
+
+        ℓ = logdensity(Exponential(2), r)
+        @test ℓ ≈ logdensity(Exponential(β=2), r)
+        @test ℓ ≈ logdensity(Exponential(λ=0.5), r)
+        @test ℓ ≈ logdensity(Exponential(logβ=log(2)), r)
+        @test ℓ ≈ logdensity(Exponential(logλ=log(0.5)), r)
+    end
+
     @testset "NegativeBinomial" begin
         D = NegativeBinomial{(:r, :p)}
         par = transform(asparams(D), randn(2))