# 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

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## Basic Examples(5)

Log10 gives the logarithm to base 10:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion shifted from the origin:

Asymptotic expansion at a singular point:

## Scope(42)

### Numerical Evaluation(6)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate efficiently at high precision:

Log10 can deal with realvalued intervals:

Log10 threads elementwise over lists and matrices:

### Specific Values(5)

Values of Log10 at fixed points:

Values at infinity:

Zero argument gives a symbolic result:

Zero of Log10:

Find a value of x for which the Log10[x]=0.5 using Solve:

### Visualization(3)

Plot the Log10 function:

Plot the real part of :

Plot the imaginary part of :

Polar plot with :

### Function Properties(10)

Log10 is defined for all positive values:

Log10 is defined for all nonzero complex values:

Log10 achieves all real values:

The range for complex values:

Log10 is not an analytic function:

Nor is it meromorphic:

Log10 has a branch cut along the negative real axis:

Log10 is monotonic on the positive reals:

Log10 is injective:

Log10 is surjective:

Log10 is neither non-negative nor non-positive:

Log10 has both singularities and discontinuities for x0:

Log10 is concave on the positive reals:

### Differentiation(3)

First derivative:

Higher derivatives:

Plot the higher derivatives:

Formula for the  derivative:

### Integration(4)

Compute the indefinite integral using Integrate:

Definite integral:

Definite integral of Log10:

More integrals:

### Series Expansions(5)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

General term in the series expansion using SeriesCoefficient:

Log10 can be applied to power series:

Asymptotic expansions at the branch cut:

### Function Identities and Simplifications(6)

Basic identity for Log10:

Logarithm of a power function simplification:

Simplify logarithms with assumptions:

Logarithm of a product:

Change of base:

Expand assuming real variables x and y:

## Applications(2)

Find the real exponent of a nonzero number:

Evaluate the transform at a point:

Plot the spectrum:

Plot both the spectrum and the plot phase using color:

Plot the spectrum in the complex plane using ParametricPlot3D:

## Properties & Relations(2)

Simplify the logarithm with assumptions:

Solve logarithmic equations: