# How to use the IMLOG10 function

The IMLOG10 function calculates the base 10 logarithm (common logarithm) of a complex number in x + yi or x + yj text format.

The letter j is used in electrical engineering to distinguish between the imaginary value and the electric current.

**Provide an example equation of when the base 10 logarithm is needed with real values ?**

The following equation can be solved using the base 10 logarithm: 10^{x}=100

becomes

log10(10^{x}) = log10(100)

becomes

x log10(10) = 2

log10(10) = 1

x*1=2

equals

x = 2

**What is the difference between the natural logarithm and the base 10 logarithm?**

The natural logarithm uses e as the base and the common logarithm uses 10 as the base.

**What is the difference between the IMLOG10 function and the IMLOG2 function?**

The IMLOG function uses the common logarithm or 10 as the base and the IMLOG2 function uses 2 as the base.

### Table of Contents

## 1. IMLOG10 Function Syntax

IMLOG10(*inumber*)

## 2. IMLOG10 Function Arguments

inumber |
Required. A complex number in x+yi or x+yj text format. |

## 3. IMLOG10 function example

The image above demonstrates a formula in cell B28 that calculates the base 10 logarithm of a complex number specified in cell B25.

Cell C28 calculates the real number from the complex number in cell B28. Cell D28 extracts the imaginary number from the complex number specified in cell B28.

The real and imaginary numbers separated in a cell each allow us to graph the complex number on the complex plane.

Formula in cell B28:

The chart above demonstrates the complex plane, the y-axis the the imaginary axis and the x-axis is the real axis.

Complex number 5+5i is the light blue line in the first quadrant. The base 10 logarithm of 5+5i is the green line also in the first quadrant.

### 3.1 Explaining formula

#### Step 1 - Populate arguments

IMLOG10(*inumber*)

becomes

IMLOG10(B25)

#### Step 2 - Evaluate the IMLOG10 function

IMLOG10(B25)

becomes

IMLOG10("5+5i")

and returns

0.849485002168009+0.34109408846046i

## 4. How is the IMLOG10 function calculated in detail?

The base 10 logarithm of a complex number is calculated like this:

C = x + yi

IMLOG10(C) =log_{10}(x+yi)=(log_{10}e)ln(x+yi)

For example, C = 5 + 5i

IMLOG10(5 + 5i) =log_{10}(5+5i)=(log_{10}e)ln(5+5i)

becomes

IMLOG10(5 + 5i) =(log_{10}e)ln(5+5i)=0.434294481903252*ln(5+5i)

becomes

IMLOG10(5 + 5i) =0.434294481903252*ln(5+5i)=0.434294481903252*(1.95601150271407+0.785398163397448i)

and returns

IMLOG10(5 + 5i) =0.849485002168009+0.34109408846046i

## 5. The IMLOG10 function not working - #NUM error

The base 10 logarithm function IMLOG10(x+yi) is not defined for x=0, so IMLOG10(0) is not a valid expression. The IMLN function will return a #NUM error if the argument is zero.

The IFERROR function can help you handle errors by returning a value of your choice if an error value occurs.

**Why is it not possible to calculate the base 10 logarithm of a complex number that has a real part of zero and an imaginary part of 0?**

The formula to calculate the base 10 logarithm of a complex number is log_{10}(x+yi)=(log_{10}e)ln(x+yi)

There is no solution to ln(0) which is why the calculation is not possible.

### Useful links

IMLOG 10 function - Microsoft

Common logarithm

Value of Log 10

### Functions in 'Engineering' category

The IMLOG10 function function is one of 42 functions in the 'Engineering' category.

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