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As its name suggests, it is the most frequently used form of logarithm. It is used, for example, in our decibel calculator. Logarithm tables that aimed at easing computation in the olden times usually presented common logarithms, too.

To demonstrate how useful it was in pre-calculator times, let's assume that you need to compute the product of 5.89 × 4.73 without any electronic device. You could do it by merely multiplying things out on paper; however, it would take a bit of time. Instead, you can use the logarithm rule with log tables and get a relatively good approximation of the result.If you want to compute a number's natural logarithm, you need to choose a base that is approximately equal to 2.718281. Conventionally this number is symbolized by e, named after Leonard Euler, who defined its value in 1731. Accordingly, the logarithm can be represented as logₑx, but traditionally it is denoted with the symbol ln(x). You might also see log(x), which also refers to the same function, especially in finance and economics. Therefore, y = logₑx = ln(x) which is equivalent to x = eʸ = exp(y). The other popular form of logarithm is the common logarithm with the base of 10, log₁₀x, which is conventionally denoted as lg(x). It is also known as the decimal logarithm, the decadic logarithm, the standard logarithm, or the Briggsian logarithm, named after Henry Briggs, an English mathematician who developed its use. You may notice that even though the frequency of compounding reaches an unusually high number, the value of (1 + r/m)ᵐ (which is the multiplier of your initial deposit) doesn't increase very much. Instead, it becomes somewhat stable: it's approaching a unique value already mentioned above, e ≈ 2.718281. lg ( 5.89 × 4.73 ) ≅ 1.4449761 \text{lg}(5.89 \times 4.73) ≅ 1.4449761 lg ( 5.89 × 4.73 ) ≅ 1.4449761 in the left form field (or top field for the mobile version of the calculator) you should see a value of 0.39370079 inches;

One practical way to understand the function of the natural logarithm is to put in the context of compound interest. That is the interest that is calculated on both the principal and the accumulated interest. We still don't know what the exact result is, so we take the exponent of both sides of the equation above with some change on the right side.

If in the left field you will be input the value in inches then automatically in the right field will be shown the corresponding value in centimeters. If you need to perform the reverse operation you must enter the value in centimeters in the right field and then automatically in the left field will be displayed the desired value in inches. lg ( 5.89 × 4.73 ) = lg ( 5.89 ) + lg ( 4.73 ) ≅ 0.770115 + 0.6748611 \text{lg}(5.89 \times 4.73) =\text{lg}(5.89) + \text{lg}(4.73) ≅ 0.770115 + 0.6748611 lg ( 5.89 × 4.73 ) = lg ( 5.89 ) + lg ( 4.73 ) ≅ 0.770115 + 0.6748611