Derivative of Logarithmic Functions
The derivative of a logarithmic function is the rate of change of the function at a specific point. For the natural logarithm function, ln(x), its derivative is 1/x. In general, the derivative of the logarithm of base b, log_b(x), is 1/(x * ln(b)). This concept is important in calculus for finding slopes and rates of change in logarithmic functions.
6 Key excerpts on "Derivative of Logarithmic Functions"
eBook - ePub- Gregory Hill, Mel Friedman(Authors)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...In symbols, this means and substituting e x for y gives The graph of f (x) = e x is shown below with the line tangent to the function at the point (0, 1). Notice that the slope of the line appears to be 1. This will be accepted without formal proof. Figure 4.1 Recall the alternate definition of a derivative, Let a = 0, replace x with h, and apply it to the function f (x) = e x. Then the slope of the tangent to the graph at x = 0 is Now examine the derivative of e x by definition. Three more key derivatives are given here without proof. If a positive constant and a ≠ 1, then If y = ln(x), then If a is a positive constant and a ≠ 1, then The last derivative given does not really need to be memorized. It can be easily established by using the change of base rule for logarithms, Since ln(a) is a constant, taking the derivative of is no different than taking the derivative of ln(x) and multiplying the result by As with the trigonometric functions, the properties of exponential and logarithmic functions will be crucial to recall, especially in multiple-choice situations where a simplified choice of the derivative may be offered, as in Example 4.11. EXAMPLE 4.11 If f (x) = log 3 (x), then f ′(2) = (A) log 3 2 (B) log 3 1 (C) (D) (E) SOLUTION It seems that the correct solution is not offered, but (D) is the proper choice. Because of the property of logarithms, This example shows the importance of knowing the properties of logarithms and exponential functions. 4.6 HIGHER-ORDER DERIVATIVES If the derivative of a function exists and is itself differentiable, it is possible to take the derivative of the derivative. Naturally, this is called the second derivative. If the first derivative represents the instantaneous rate of change of the function at a point, the second derivative represents the rate of change of the rate of change. The significance of this idea will be explored in the next chapter...
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...With regard to the relative error, one should divide both sides of Eq. (10.1) by y (or f, for that matter), according to (10.14) and then multiply and divide the right‐hand side by x, viz. (10.15) As will be proven later, the derivative of a logarithm is given by the derivative of its argument divided by the argument itself, so Eq. (10.15) can be rephrased as (10.16) hence, the relative error dx / x of the independent variable echoes upon dy / y via the logarithmic derivative, i.e. d ln f / d ln x. One may finally revisit Eq. (10.1) as (10.17) encompassing a composite function Φ of f { x } rather than f { x } itself, and resorting to chain differentiation (to be addressed shortly) – where cancellation of dx between numerator and denominator produces (10.18) in other words, Φ { f } may, for the sake of differentiation, be treated as a function of variable f (despite f being a function of x). By the same token, Eq. (10.7) suggests (10.19) where Ψ denotes any set of algebraic operations upon univariate functions f 1 { x }, f 2 { x }, …, f n { x } – which leads to (10.20) again after dropping dx from both numerator and denominator. Therefore, the differential of any function of functions may be calculated via direct application of the rules of differentiation, as long as the differentials of the corresponding constitutive functions are made available. 10.2 Derivative 10.2.1 Definition The derivative of a function f { x }, at a given point x 0, is defined as (10.21) graphically speaking, this corresponds to the limit of a straight line secant to the plot of f { x } at points with coordinates (x 0,) and (x 0 + h,), as emphasized in Fig. 10.2...
- Craig Adam(Author)
- 2011(Publication Date)
- Wiley(Publisher)
...Similarly: f (x) = (e x + e −2 x) 2 = e 2 x + 2 e x e −2 x + e −4 x = e 2 x + 2 e − x + e −4 x. Self-assessment exercises 1. If y = e x +2 and z = e −2 x evaluate and simplify the following: (a) f (x) = yz (b) (c) (d) 2. Multiply out and simplify: (a) f (x) = (1 + e 2 x)(e x − e − x) (b) f (x) = e −2 x (1 − e x) 2 3.2 Origin and definition of the logarithmic function If you understand the principles behind the exponential function, the logarithmic function should seem straightforward. There is however a slight complication resulting from the common use of different bases for logarithms. The inverse of the exponential function is the logarithmic function – or more correctly the natural logarithmic function, denoted Ln or ℓn or sometimes log e. This function answers the question “To what power should e be raised to generate the required number, y ?”. Thus: Unlike the exponential function which changes rapidly, the logarithmic function is a slowly changing function and so may be used to compress data that has a wide dynamic range. The most significant example of this is the pH scale for the measurement of H 3 O + ion concentration (acidity), which will be discussed in detail later. Another example is the decibel or dB scale, which is a logarithmic scale, to the base 10, used to measure sound intensity levels. On these scales, intensities that differ by a factor of 10 are expressed as having a pH or dB difference of 1, as this represents the difference in their exponents when written to this base. The graph of y = Ln(x) is illustrated in Figure 3.2. Note that, since A 0 = 1 for all values of A, then Ln(1) = 0, and numbers less than unity have a negative logarithm...
eBook - ePub- Arsen Melkumian(Author)
- 2012(Publication Date)
- Routledge(Publisher)
...3 Exponential and logarithmic functions Logarithmic functions are indispensable in economic analysis as they can transform multiplicative relationships between economic variables into additive ones. In addition, economists often choose (for the sake of convenience) to optimize the natural log of an objective function instead of the objective function itself. Exponential functions are very useful when modeling the growth of a certain economic variable. For example, we can use exponential functions to model the growth of the population of a country. This chapter introduces logarithmic and exponential functions and closes with some Mathematica examples. 3.1 Logarithmic function Consider the following equation: In the equation above the base is equal to 3, and the exponent is equal to 5. The power to which 3 must be raised to yield 243 is called the logarithm (or log) to the base 3 of 243. So, logarithm to the base 3 of 243 is equal to 5: In general, if where a, B > 0 and a ≠ 1, and x ∈ ℝ, then we can write that For example, 2 5 = 32 implies that log to the base 2 of 32 equals 5 E XAMPLE 3.1 (a) log 2 1 = 0, since 2 raised to the power of 0 is 1. (b) log 7 7 = 1, since 7 raised to the power of 1 is 7. (c) log 10 = −1, since 10 raised to the power of −1 is. (d) log 3 81 = 4, since 3 raised to the power of 4 is 81. (e) log 5 (−25) is not defined, since −25 < 0. (f) log 12 1728 = 3, since 12 raised to the power of 3 is 1728. (g) log 5 = −2, since 5 needs to be raised to the power of −2 in order to get. Often economists work with logarithms to the base e, where e is the irrational number 2.718 … known as the exponential. Logarithms to the base e are referred to as natural logarithms. We can write either log e B or ln B to refer to the natural logarithm of B. Now, consider the function y = f (x) = ln x. The graph of y is given in Figure 3.1...
eBook - ePub- Alan J. Cann(Author)
- 2013(Publication Date)
- Wiley(Publisher)
...Rule 1 – log a (x * y)=log a (x) + log a (y), i.e. you can multiply two numbers by adding their logarithms. 2. Rule 2 – log a (x/y) = log a (x) – log a (y), i.e. you can divide two numbers by subtracting their logarithms. 3. Rule 3 – log a (x) n = n log a (x), i.e. the logarithm of an exponent gives the original number. Remember the exponential function y = z x described in Section 6.2? When this function is plotted on semi-log graph paper (also called loglinear graph paper, where the y axis has a log rather than a linear scale), or if the log values of y are plotted, the log transformation of the exponential function results in the exponential curve being transformed into a straight line (Figure 6.2). Logarithms can be used to solve exponential equations which occur commonly in biology, e.g. pH: Figure 6.2 The log transformation of the exponential function results in the exponential curve being transformed into a straight line Human blood plasma has a typical H + concentration (written as ‘[H + ]’) of 10 −7.4 M. Therefore: Another example is the exponential growth of populations. The growth of a population is described by the equation: where N is the size of the population at time t; N 0 is the size of the original population; e is Euler’s number; and λ is the growth rate (as a decimal). During the growth phase of a bacterial culture, the rate of increase of cells is proportional to the number of bacteria present. The constant of proportionality, μ, is an index of the growth rate and is called the growth rate constant: The value of μ can be determined from the following equation: The natural logarithm of the number of cells at time t minus the natural logarithm of the number of cells at time zero (t 0) equals the growth rate constant multiplied by the time interval. For most purposes, it is easier to use log 10 values rather than natural logarithms...
eBook - ePub- R. Stafford Johnson(Author)
- 2013(Publication Date)
- Bloomberg Press(Publisher)
...Appendix B Uses of Exponents and Logarithms Exponential Functions An exponential function is one whose independent variable is an exponent. For example: where: y = dependent variable t = independent variable b = base (b > 1) In calculus, many exponential functions use as their base the irrational number 2.71828, denoted by the symbol e: An exponential function that uses e as its base is defined as a natural exponential function. For example: These functions also can be expressed as: In calculus, natural exponential functions have the useful property of being their own derivative. In addition to this mathematical property, e also has a finance meaning. Specifically, e is equal to the future value (FV) of $1 compounded continuously for one period at a nominal interest rate (R) of 100 percent. To see e as a future value, consider the future value of an investment of A dollars invested at an annual nominal rate of R for t years, and compounded m times per year. That is: (B.1) If we let A = $1, t = one year, and R = 100 percent, then the FV would be: (B.2) If the investment is compounded one time (m = 1), then the value of the $1 at end of the year will be $2; if it is compounded twice (m = 2), the end-of-year value will be $2.25; if it is compounded 100 times (m = 100), then the value will be 2.7048138. As m becomes large, the FV approaches the value of $2.71828. Thus, in the limit: (B.3) If A dollars are invested instead of $1, and the investment is made for t years instead of one year, then given a 100 percent interest rate the future value after t years would be: (B.4) Finally, if the nominal interest rate is different than 100 percent, then the FV is: (B.5) To prove Equation (B.5), rewrite Equation (B.1) as follows: (B.6) If we invert R/m in the inner term, we get: (B.7) The inner term takes the same form as Equation (B.2). As shown earlier, this term, in turn, approaches e as m approaches infinity...
