## Natural Log (Ln) calculator

Calculate the natural logarithm of X to the base of e as **log _{e}X**:

#### Ln Calculator

Result:

Value y = log_{e}X: |
0 |

### What is Natural Logarithm?

Logarithm (or log) is a mathematical operation which is the inverse of exponentiation. This means that, the logarithm of a number is the power that the base must be raised to, to obtain this number. The exponentiation is written as π₯ = b^{y} . The corresponding log notation is π¦ = log_{b} π₯. If the base is e, the notation is π₯ = e^{y} , and the log notation is π¦ = log_{e} π₯ or π¦ = ln π₯.

There are two types of logarithm: the common logarithm and natural log. In usual notations, the word βlogβ denotes the common logarithm where the base is 10, although any base can be used. For example, log_{10} 5 is simply written as log 5. For any other number, the base is always mentioned.

The other kind of logarithm is natural log, where the base is e, which is Eulerβs number. This is an irrational number and it is equal to 2.7182. In this case, the natural log is denoted by ln or log with e as the base. For example, log_{e} 5 = ln 5. The natural log defines how many times e needs to be multiplied with itself to get 5.

The main difference between the common logarithm and natural logarithm is the base.

### Differences between log and natural log

The difference between log and ln is crucial for solving mathematical problems using these functions. Some of the main differences between these two functions have been listed below.

log | ln |
---|---|

log alone indicates a logarithm with base 10, even though other bases can be used. For any other base, it is mentioned explicitly, but for 10, this may be omitted. | ln refers to logarithms with base e. |

This is written as π₯ = b^{y} in exponent form and π¦ = log_{b} π₯ using the log notation. If the base is 10, then π₯ = 10^{y} and π¦ = log_{10} π₯, or simply y = log x. |
When the base is e, the notation is π₯ = e^{y} , and the log notation is π¦ = log_{e} π₯ or π¦ = ln π₯. |

The log function is called the common logarithm. | ln is called the natural logarithm. |

The log function indicates how many times the base must be multiplied with itself to get a particular number, or what is the power that the base must be raised to get a particular number. For example, for the base 10, log (100) = 2, which implies that 10^{2} = 100. |
The log function indicates how many times βeβ must be multiplied with itself to get a particular number, or what is the power that βeβ must be raised to get a particular number. For example, ln (7.38) β 2 and e^{2} = 7.38 |

### Natural log formula

The formula for natural log is π¦ = log_{e} π₯ = ln π₯, which is equivalent to π₯ = e^{y}

The ln function and e are inverse of each other. Therefore, log_{e} (e^{x} ) = ln e^{y} = π₯, and e^{ln (x)} = π₯

The natural log of 0 and any negative number is not defined. The natural log of 1 is 0, that is ln 1 = 0. The natural log of e is 1, that is, log_{e} π = ln π = 1.

### Natural log rules

Natural logs follow several rules which have been listed below.

rule | example |
---|---|

Product rule: $$ ln(x*y) = ln\,x \;+\; ln,y $$ | For example, $$ ln(4*8) = ln\,4 \;+\; ln\,8 = 3.4657 $$ |

Quotient rule: $$ ln({x\over y}) = ln\,x \;-\; ln,y $$ | For example, $$ ln({4\over8}) = ln\,4 \;-\; ln\,8 = β0.69315 $$ |

Power rule: $$ ln(x^y) = y\;ln\,x $$ | For example, $$ ln(5^2) = 2\;ln\,5 = β0.69315 $$ |

Reciprocal rule: $$ ln({1\over x}) = -\,ln\,x $$ | For example, $$ ln({1\over3}) = -\,ln\,3 $$ |

$$ log_e (e^x ) = ln \; e^x = π₯ $$ | |

$$ e^{ln(x)} = x$$ | |

The natural log of a negative number is undefined. | |

The natural log of 0 is undefined. | |

The natural log of e is 1, that is, log_{e} π = ln π = 1. |

### Areas of application

Natural log and common log have the same areas of application. Both are very convenient ways of expressing large numbers. In exponential functions involving growth and decay, the natural log function is more commonly used. The natural log helps us to compute the time taken for growth and decay.

Given below are some example problems involving natural log.

__Question:__ Calculate ln βπ

__Answer:__ In this problem, ln βπ = ln e^{1/2} = (1/2) ln e = 1/2

__Question:__ Solve for x, ln (5x-7) = 3.

__Answer:__ Here, ln (5x-7) = 3.

When there are multiple terms in the ln parentheses, it helps to explicitly write e as the base and all other terms as exponents of e. In doing so, we get the form e^{ln (x)} = π₯, which aids in the simplification process.

Therefore, the given equation can be written as e^{ ln (5x-7)} = e^{3}.

As e^{ln (x)} = π₯, the above step reduces to 5π₯ β 7 = e^{3}.

Since e is a constant, its powers can be found easily using a calculator. Hence, we obtain

5π₯ β 7 = 20.0855

5π₯ = 27.0855

π₯ = 5.4171

__Question:__ The half-life of caffeine in the human body is close to 6 hours. If John drinks a cup of coffee at 8 am in the morning, what is the amount of caffeine left in his system at 5 pm?

__Answer:__ In situations involving half-life, exponential decay takes place. The half-life indicates the time taken by a value to reduce to half its initial value with exponential decay. This is commonly used for radioactive substances.

In the current problem, we start with the formula for exponential decay, π₯(π‘) = x(0)π^{kt}, where x(0) is the initial amount of caffeine in the system, t is the time in hours, and k is the growth rate.

First, we need to calculate k using the information provided by the half-life. Hence, x(0) = 1 cup of coffee, t = 6 hours and x(t) = Β½.

$$ x(t) = x(0)e^{kt} $$

$$ {1\over2} = 1*e^{6k} \; or \; {1\over2} =e^{6k} $$

Taking the natural log on both sides gives,

$$ ln {1\over2} = ln \;e^{6k} $$

As e^{ln (x)} = π₯, the above step reduces to

$$ ln {1\over2} = 6k$$

Hence,

$$ {{ln {1\over2}}\over 6} = k = β0.1155 $$

The negative sign indicates exponential decay. Now, the time difference between 8 am and 5 pm is required to find the amount of caffeine present at 5 pm. Hence, t = 9.

$$ x(9) = x(0)e^{kt} $$

$$ x(9) = 1 * e^{(-0.1155*9)} = e^{-1.0395} = 0.35 $$

Therefore, at 5pm, the amount of caffeine in the system is 0.35 times the original amount.