Log10
Log10[x]
gives the base-10 logarithm of x.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- Log10 gives exact rational number results when possible.
- For certain special arguments, Log10 automatically evaluates to exact values.
- Log10 can be evaluated to arbitrary numerical precision.
- Log10 automatically threads over lists.
Examples
open all close allBasic Examples (5)
Log10 gives the logarithm to base 10:
Log10[1000000]Log10[2.]Log10[x]Plot over a subset of the reals:
Plot[Log10[x], {x, 0, 5}]Plot over a subset of the complexes:
ComplexPlot3D[Log10[z ^ 3], {z, -2 - 2I, 2 + 2I}, PlotLegends -> Automatic]Series expansion shifted from the origin:
Series[Log10[1 + x], {x, 0, 5}]Asymptotic expansion at a singular point:
Series[Log10[x], {x, -1, 3}, Assumptions -> Im[x] > 0]Scope (44)
Numerical Evaluation (8)
Log10[5.]N[Log10[45], 100]The precision of the output tracks the precision of the input:
Log10[45.211111111111111111]N[Log10[4 + I]]Evaluate efficiently at high precision:
Log10[71`100]//TimingLog10[8956`100000];//TimingLog10 can deal with real‐valued intervals:
Log10[Interval[{1 / 3, 2}]]Log10 threads elementwise over lists and matrices:
Log10[{1, 10, 100, 1000}]Log10[(| | |
| :- | :- |
| 1 | u |
| v | 0 |)]//MatrixFormCompute average-case statistical intervals using Around:
Log10[ Around[1 / 101, 0.01]]Compute the elementwise values of an array:
Log10[{{1, 100}, {0, 1 / 100}}]Or compute the matrix Log10 function using MatrixFunction:
MatrixFunction[Log10, {{1, 100}, {0, 1 / 100}}]//FullSimplifySpecific Values (5)
Values of Log10 at fixed points:
Table[Log10[n ], {n, {0, 1, 10, 100}}]Table[Log10[1 / n ], {n, {1, 10, 100}}]Log10[ComplexInfinity]Log10[Infinity]Zero argument gives a symbolic result:
Log10[0]Zero of Log10:
Log10[1]Find a value of x for which the Log10[x]=0.5 using Solve:
xval = x /. FindRoot[Log10[x] == 0.5, {x, 0.1}]Plot[Log10[x], {x, 0, 2 E}, Epilog -> Style[Point[{xval, Log10[xval]}], PointSize[Large], Red]]Visualization (3)
Plot the Log10 function:
Plot[Log10[x], {x, 0, 4}]
ComplexContourPlot[Re[Log10[z]], {z, -2 - 2I, 2 + 2 I}, Contours -> 20]
ComplexContourPlot[Im[Log10[z]], {z, -2 - 2I, 2 + 2 I}, Contours -> 20]
PolarPlot[Log10[ϕ], {ϕ, 0, 12π}, Frame -> True]Function Properties (10)
Log10 is defined for all positive values:
FunctionDomain[Log10[x], x]Log10 is defined for all nonzero complex values:
FunctionDomain[Log10[x], x, Complexes]Log10 achieves all real values:
FunctionRange[Log10[x], x, y]FunctionRange[Log10[x], x, y, Complexes]Log10 is not an analytic function:
FunctionAnalytic[Log10[x], x]FunctionMeromorphic[Log10[x], x]Log10 has a branch cut along the negative real axis:
Limit[Log10[-10 + ϵ I], ϵ -> 0, Direction -> "FromAbove"]Limit[Log10[-10 + ϵ I], ϵ -> 0, Direction -> "FromBelow"]Log10 is monotonic on the positive reals:
FunctionMonotonicity[Log10[x], x, PositiveReals]Log10 is injective:
FunctionInjective[Log10[x], x]Plot[{Log10[x], 1 / 2}, {x, 0, 5}]Log10 is surjective:
FunctionSurjective[Log10[x], x]Plot[{Log10[x], 2}, {x, 0, 150}]Log10 is neither non-negative nor non-positive:
FunctionSign[Log10[x], x]Log10 has both singularities and discontinuities for x≤0:
FunctionSingularities[Log10[x], x]FunctionDiscontinuities[Log10[x], x]Log10 is concave on the positive reals:
FunctionConvexity[{Log10[x], x > 0}, x]TraditionalForm formatting:
Log10[n]//HoldForm//TraditionalFormDifferentiation (3)
D[Log10[x], x]Table[D[Log10[x], {x, k}], {k, 1, 5}]Plot[%, {x, -10, 10}, PlotLegends -> {"First Derivative", "Second Derivative", "Third Derivative", "Fourth Derivative", "Fifth Derivative"}]
derivative:D[Log10[x], {x, k}]Integration (4)
Compute the indefinite integral using Integrate:
Integrate[Log10[x], x]Integrate[Log10[x], {x, 0, 1}]Definite integral of Log10:
Integrate[Log10[x], {x, 0, 1}]Integrate[Log10[x]Log2[x], x]//FullSimplifyIntegrate[Log10[z] ^ a, z]Integrate[t ^ (α - 1)Log10[t], {t, 0, 1}, Assumptions -> Re[α] > 0]Series Expansions (5)
Find the Taylor expansion using Series:
Series[Log10[x], {x, 2, 4}]Plots of the first three approximations around 
:
terms = Normal@Table[Series[Log10[x], {x, 1, m}], {m, 1, 5, 2}];
Plot[{Log10[x], terms}, {x, -5, 5}, PlotRange -> {-2, 2}]General term in the series expansion using SeriesCoefficient:
SeriesCoefficient[Log10[x], {x, 1, n}]FourierSeries[Log10[x], x, 1]// FullSimplifyLog10 can be applied to power series:
Log10[x + (x^2/2) + (x^3/3) + O[x]^4]Asymptotic expansions at the branch cut:
Series[Log10[x], {x, -2, 3}]Function Identities and Simplifications (6)
Basic identity for Log10:
10 ^ Log10[x]Logarithm of a power function simplification:
FunctionExpand[Log10[x^a], x > 0 && a∈Reals]Simplify logarithms with assumptions:
Log10[10 ^ n]Simplify[%, n > 0]FunctionExpand[Log10[x y], x > 0 && x + y ≥ 0]FullSimplify[Log10[x] == (Log[a, x]/Log[a, 10])]Expand assuming real variables x and y:
ComplexExpand[Log10[x + I y]]//FullSimplifyApplications (2)
Find the real exponent of a nonzero number:
Log10[1.75]RealExponent[1.75]Evaluate the transform at a point:
F = ZTransform[Sin[n 2Pi / 3](2 / 3) ^ n, n, Exp[I ω]]Plot[Log10[Abs[F] ^ 2], {ω, 0, 2π}, Ticks -> {{0, π, 2π}, Automatic}, AxesOrigin -> {0, -1.2}]Plot both the spectrum and the plot phase using color:
Plot[Log10[Abs[F] ^ 2], {ω, 0, 2π}, Ticks -> {{0, π, 2π}, Automatic}, ColorFunction -> Function[ω, Evaluate@Hue[Arg[F] / (2Pi) + 1 / 2]], ColorFunctionScaling -> False, Filling -> Axis, AxesOrigin -> {0, -1.2}]Plot the spectrum in the complex plane using ParametricPlot3D:
ParametricPlot3D[{Cos[ω], Sin[ω], Log10[Abs[F] ^ 2]}, {ω, 0, 2π}, BoxRatios -> {1, 1, 1}]
MathWorld
An Elementary Introduction to the Wolfram LanguageText
Wolfram Research (2008), Log10, Wolfram Language function, https://reference.wolfram.com/language/ref/Log10.html.
CMS
Wolfram Language. 2008. "Log10." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Log10.html.
APA
Wolfram Language. (2008). Log10. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Log10.html