Semra Özal CONNECTIONS OF LOGARITHMIC FUNCTIONS Logarithm and exponential functions have close relationship and they are inverse function of each other in a way. Before explicitly clarifying this inverse relationship, we should analyze their definitions. Logarithm means, in mathematics, “The exponent that indicates the power to which a base number is raised to produce a given number “2 Exponential function means that “mathematical function in which an independent variable appears in one of the exponents —called also exponential”3.We told that they have inverse relation between each other.
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To give an another definition of complex number, we have to show z= ew for w, where z is any non-zero complex number. If we assume w = u + iv then eueiv = |z|e^i arg z and this means that
|z| = e^u , v = arg z .
The equation |z| = e^u is a real equation, so we can write u = ln |z|, where ln |z| is the ordinary logarithm with positive real numbers. Hence, w = u + iv = ln |z| + i arg z = ln |z| + i(Arg z + 2n) , n = 0 , ±1 , ±2 , ±3 , . . .6 We profoundly examined the connection between exponential and logarithmic functions. However logarithmic functions are associated with many other areas of science. We use logarithm in such that the decibel scale for the loudness of sound, the Richter scale of earthquake magnitudes, and the astronomical scale of stellar brightness are all logarithmic scales. When we want to calculate radioactive decay, we use logarithmic functions. We know that radioactive atoms decay randomly.” If we have a sample of atoms, and we specify a time interval short enough that the population of atoms has not changed significantly through decay, then the proportion of atoms decaying in our short time interval will be proportional to the length of the interval. “ 7 In the light of this information we get an expression for the number of atoms remaining, N, as a proportion of the number of atoms N0 at time 0, in terms of time, …show more content…
For example, to calculate GDP for a particular year and next year, we need to take logarithm to find implicit growth rate. Calculation of richter scale and decibel need logarithm too. Intensity of earthquake has a scale with small range such as from 1 to 10. Decibels are similar to richter scale. Sounds could be so quiet or extremely loud. Logarithms help us classification of events such as intensity of earthquakes, radioactive decay and the scale of decibels and make understanding and perception of some huge numbers
To give an another definition of complex number, we have to show z= ew for w, where z is any non-zero complex number. If we assume w = u + iv then eueiv = |z|e^i arg z and this means that
|z| = e^u , v = arg z .
The equation |z| = e^u is a real equation, so we can write u = ln |z|, where ln |z| is the ordinary logarithm with positive real numbers. Hence, w = u + iv = ln |z| + i arg z = ln |z| + i(Arg z + 2n) , n = 0 , ±1 , ±2 , ±3 , . . .6 We profoundly examined the connection between exponential and logarithmic functions. However logarithmic functions are associated with many other areas of science. We use logarithm in such that the decibel scale for the loudness of sound, the Richter scale of earthquake magnitudes, and the astronomical scale of stellar brightness are all logarithmic scales. When we want to calculate radioactive decay, we use logarithmic functions. We know that radioactive atoms decay randomly.” If we have a sample of atoms, and we specify a time interval short enough that the population of atoms has not changed significantly through decay, then the proportion of atoms decaying in our short time interval will be proportional to the length of the interval. “ 7 In the light of this information we get an expression for the number of atoms remaining, N, as a proportion of the number of atoms N0 at time 0, in terms of time, …show more content…
For example, to calculate GDP for a particular year and next year, we need to take logarithm to find implicit growth rate. Calculation of richter scale and decibel need logarithm too. Intensity of earthquake has a scale with small range such as from 1 to 10. Decibels are similar to richter scale. Sounds could be so quiet or extremely loud. Logarithms help us classification of events such as intensity of earthquakes, radioactive decay and the scale of decibels and make understanding and perception of some huge numbers