Logarithm Calculator. Step-by-step solution

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Last update: 2024-06-18

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Logarithm of...

Base

Result:

1.0986122886681

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 = (3 - 1) ÷ (3 + 1)

(3 - 1) ÷ (3 + 1)

3 - 1

= 2

2 ÷ (3 + 1)

3 + 1

= 4

2 ÷ 4

2 ÷ 4

= 0.5

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: 0.5
i: Iteration number 1, 2, 3, ..., n
Initial result: 0
Initial term: 0

Term = 0.5^(1 × 2 - 1) ÷ (1 × 2 - 1) = 0.5
Result = 0 + 0.5 = 0.5

= 0.5

Term = 0.5^(2 × 2 - 1) ÷ (2 × 2 - 1) = 0.041666666666667
Result = 0.5 + 0.041666666666667 = 0.54166666666667

= 0.54166666666667

Term = 0.5^(3 × 2 - 1) ÷ (3 × 2 - 1) = 0.00625
Result = 0.54166666666667 + 0.00625 = 0.54791666666667

= 0.54791666666667

...

Relative change: 2.0211370946362E-16 < Dynamic Epsilon: 1.0E-15 = Stopping calculations

= 0.54930614433405

Iterations: 24

= 0.54930614433405

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

0.54930614433405 ÷ 0.49999999999999

= 1.0986122886681

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