Log

log(b,x)

Compute the base b logarithm of x. Throws DomainError for negative Real arguments.

Examples

julia> log(4,8)
1.5

julia> log(4,2)
0.5

julia> log(-2, 3)
ERROR: DomainError with -2.0:
log was called with a negative real argument but will only return a complex result if called with a complex argument. Try log(Complex(x)).
Stacktrace:
 [1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31
[...]

julia> log(2, -3)
ERROR: DomainError with -3.0:
log was called with a negative real argument but will only return a complex result if called with a complex argument. Try log(Complex(x)).
Stacktrace:
 [1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31
[...]

julia> log(100,1000000)
2.9999999999999996

julia> log10(1000000)/2
3.0

log(x)

Compute the natural logarithm of x.

Throws DomainError for negative Real arguments. Use complex arguments to obtain complex results. Has a branch cut along the negative real axis, for which -0.0im is taken to be below the axis.

See also ℯ, log1p, log2, log10.

Examples

julia> log(2)
0.6931471805599453

julia> log(-3)
ERROR: DomainError with -3.0:
log was called with a negative real argument but will only return a complex result if called with a complex argument. Try log(Complex(x)).
Stacktrace:
 [1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31
[...]

julia> log(-3 + 0im)
1.0986122886681098 + 3.141592653589793im

julia> log(-3 - 0.0im)
1.0986122886681098 - 3.141592653589793im

julia> log.(exp.(-1:1))
3-element Vector{Float64}:
 -1.0
  0.0
  1.0

log(A::AbstractMatrix)

If A has no negative real eigenvalue, compute the principal matrix logarithm of A, i.e. the unique matrix $X$ such that $e^X = A$ and $-\pi < Im(\lambda) < \pi$ for all the eigenvalues $\lambda$ of $X$. If A has nonpositive eigenvalues, a nonprincipal matrix function is returned whenever possible.

If A is symmetric or Hermitian, its eigendecomposition (eigen) is used, if A is triangular an improved version of the inverse scaling and squaring method is employed (see ^[AH12] and ^[AHR13]). If A is real with no negative eigenvalues, then the real Schur form is computed. Otherwise, the complex Schur form is computed. Then the upper (quasi-)triangular algorithm in ^[AHR13] is used on the upper (quasi-)triangular factor.

Examples

julia> A = Matrix(2.7182818*I, 2, 2)
2×2 Matrix{Float64}:
 2.71828  0.0
 0.0      2.71828

julia> log(A)
2×2 Matrix{Float64}:
 1.0  0.0
 0.0  1.0