Logarithm
![The graph of the logarithm to base 2 crosses the x axis (horizontal axis) at 1 and passes through the points with coordinates (2, 1), (4, 2), and (8, 3). For example, log2(8) = 3, because 23 = 8. The graph gets arbitrarily close to the y axis, but does not meet or intersect it.](/Images/godic/202412/23/Binary_logarithm_plot_with_ticks.svg2806.png")
![Logarithm in base 2-10 of numbers 1-10. Line labels = number. X axis = base. Y axis = logarithm.](/Images/godic/202412/23/Logarithm_chart.svg2806.png")
![The 1797 Encyclopædia Britannica explanation of logarithms](/Images/godic/202412/23/Logarithms_Britannica_17972806.png")
![Schematic depiction of a slide rule. Starting from 2 on the lower scale, add the distance to 3 on the upper scale to reach the product 6. The slide rule works because it is marked such that the distance from 1 to x is proportional to the logarithm of x.](/Images/godic/202412/23/Slide_rule_example2_with_labels.svg2806.png")
In mathematics, the logarithm is the inverse operation to exponentiation. That means the logarithm of a number is the exponent to which another fixed value, the base, must be raised to produce that number. In simple cases the logarithm counts repeated multiplication. For example, the base 10 logarithm of 1000 is 3, as 10 to the power 3 is 1000 (1000 = 10 × 10 × 10 = 10); the multiplication is repeated three times. More generally, exponentiation allows any positive real number to be raised to any real power, always producing a positive result, so the logarithm can be calculated for any two positive real numbers b and x where b is not equal to 1. The logarithm of x to base b, denoted logb(x), is the unique real number y such that