Logarithm Calculator. Step-by-step solution

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Last update: 2024-06-18

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Logarithm of...

Base

Result:

2.7080502011023

Solution steps

Calculate the logarithm of the base. Variable logBase

x =

\frac{base - 1}{base + 1} ​

x = (2.718281828459 - 1) ÷ (2.718281828459 + 1)

(2.718281828459 - 1) ÷ (2.718281828459 + 1)

2.718281828459 - 1

= 1.718281828459

1.718281828459 ÷ (2.718281828459 + 1)

2.718281828459 + 1

= 3.718281828459

1.718281828459 ÷ 3.718281828459

1.718281828459 ÷ 3.718281828459

= 0.46211715726

Use Taylor series:

f (x) = \sum_{n = 0}^{\infty} (\frac{f^{(n)} (a)}{n!} ​) {(x - a)}^{n}

Where:
Function:

f (x, i) = \frac{x^{i \times 2 - 1}}{i \times 2 - 1} ​

x: 0.46211715726
i: Iteration number 1, 2, 3, ..., n
Initial result: 0
Initial term: 0

Term = 0.46211715726^(1 × 2 - 1) ÷ (1 × 2 - 1) = 0.46211715726
Result = 0 + 0.46211715726 = 0.46211715726

= 0.46211715726

Term = 0.46211715726^(2 × 2 - 1) ÷ (2 × 2 - 1) = 0.03289538885607
Result = 0.46211715726 + 0.03289538885607 = 0.49501254611607

= 0.49501254611607

Term = 0.46211715726^(3 × 2 - 1) ÷ (3 × 2 - 1) = 0.0042149309191085
Result = 0.49501254611607 + 0.0042149309191085 = 0.49922747703518

= 0.49922747703518

...

Relative change: 8.8817841970015E-16 < Dynamic Epsilon: 1.0E-15 = Stopping calculations

= 0.49999999999999

Iterations: 21

= 0.49999999999999

Calculate the logarithm of the x. Variable logX

x =

\frac{x - 1}{x + 1} ​

x = (15 - 1) ÷ (15 + 1)

(15 - 1) ÷ (15 + 1)

15 - 1

= 14

14 ÷ (15 + 1)

15 + 1

= 16

14 ÷ 16

14 ÷ 16

= 0.875

Use Taylor series:

f (x) = \sum_{n = 0}^{\infty} (\frac{f^{(n)} (a)}{n!} ​) {(x - a)}^{n}

Where:
Function:

f (x, i) = \frac{x^{i \times 2 - 1}}{i \times 2 - 1} ​

x: 0.875
i: Iteration number 1, 2, 3, ..., n
Initial result: 0
Initial term: 0

Term = 0.875^(1 × 2 - 1) ÷ (1 × 2 - 1) = 0.875
Result = 0 + 0.875 = 0.875

= 0.875

Term = 0.875^(2 × 2 - 1) ÷ (2 × 2 - 1) = 0.22330729166667
Result = 0.875 + 0.22330729166667 = 1.0983072916667

= 1.0983072916667

Term = 0.875^(3 × 2 - 1) ÷ (3 × 2 - 1) = 0.10258178710938
Result = 1.0983072916667 + 0.10258178710938 = 1.200889078776

= 1.200889078776

...

Relative change: 0 < Dynamic Epsilon: 1.0E-16 = Stopping calculations

= 1.3540251005511

Iterations: 118+

= 1.3540251005511

{log}_{e} (3) = \frac{logX}{logBase} ​

1.3540251005511 ÷ 0.49999999999999

= 2.7080502011023

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