Logarithm Basics: Definitions and Examples

Logarithm Basics: Definitions, Formulas, & Examples

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    Introduction:

    Logarithms have been an essential part of mathematical calculations for centuries. The logarithm is the inverse function of the exponential function, which is commonly used in mathematics and science. In particular, the common logarithm, also known as the base-10 logarithm, is a widely used logarithm that is frequently encountered in scientific and engineering applications. In this article, we will define what the common logarithm is, provide some examples of how to use it, and give a quiz to test your understanding.

    Definitions:

    The common logarithm of a number is defined as the power to which 10 must be raised to obtain that number. In other words, if we have a positive number “x,” the common logarithm of “x” is the power “n” to which 10 must be raised to give “x,” i.e., log(x) = n if 10^n = x. Mathematically, we can write the common logarithm as:

    log(x) = n

    where x is the positive number whose logarithm we are trying to find and n is the power to which 10 must be raised to obtain x.

    The common logarithm is denoted by the symbol “log” without a subscript. It is assumed that any use of the term “log” refers to the common logarithm unless otherwise specified.

    The logarithm is a useful tool for simplifying mathematical calculations that involve very large or very small numbers. By using the logarithm, we can convert multiplication and division operations into addition and subtraction operations, respectively, which makes calculations much simpler.

    Examples:

    1. Find the common logarithm of 100. Solution: We know that 10^2 = 100, so the common logarithm of 100 is 2. Therefore, log(100) = 2.
    2. Find the common logarithm of 1,000,000. Solution: We know that 10^6 = 1,000,000, so the common logarithm of 1,000,000 is 6. Therefore, log(1,000,000) = 6.
    3. Find the common logarithm of 1. Solution: We know that 10^0 = 1, so the common logarithm of 1 is 0. Therefore, log(1) = 0.
    4. Find the common logarithm of 0.1. Solution: We know that 10^-1 = 0.1, so the common logarithm of 0.1 is -1. Therefore, log(0.1) = -1.
    5. Find the value of x if log(x) = 3. Solution: We know that 10^3 = 1,000, so x = 1,000. Therefore, log(1,000) = 3.

    Quiz

    1. What is the definition of a logarithm?
    2. What is the base of a logarithm?
    3. What is the inverse of a logarithm?
    4. What is the logarithm of 1?
    5. What is the logarithm of 0?
    6. What is the logarithm of a negative number?
    7. What is the relationship between exponential functions and logarithmic functions?
    8. What is the change of base formula?
    9. What is the logarithmic identity for multiplication?
    10. What is the logarithmic identity for division?

     

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    Logarithm Basics:

    Alternative representations

    log(x) = log(e, x)

    log(x) = log(a) log(a, x)

    log(x) = -Li_1(1 - x)

    log(x) = -S_(0, 1)(1 - x)

    log(x) = 2 coth^(-1)((x + 1)/(x - 1)) + 2 π i for x<-1

    log(x) = 2 tanh^(-1)((x - 1)/(x + 1)) + 2 π i for x<-1

    log(x) = 2 π i - sech^(-1)((2 x)/(1 + x^2)) for -1<x<0

    log(x) = 2 i cot^(-1)((i (x + 1))/(x - 1)) + 2 π i for x<-1

    Series representations

    log(x) = - sum_(k=1)^∞ ((-1)^k (-1 + x)^k)/k for abs(-1 + x)<1

    log(x) = log(-1 + x) - sum_(k=1)^∞ ((-1)^k (-1 + x)^(-k))/k for abs(-1 + x)>1

    log(x) = 2 i π floor(arg(x - ξ)/(2 π)) + log(ξ) - sum_(k=1)^∞ ((-1)^k (x - ξ)^k ξ^(-k))/k for ξ<0

    log(x) = floor(arg(x - z_0)/(2 π)) log(1/z_0) + log(z_0) + floor(arg(x - z_0)/(2 π)) log(z_0) - sum_(k=1)^∞ ((-1)^k (x - z_0)^k z_0^(-k))/k

    log(x) = 2 i π floor((π - arg(x/z_0) - arg(z_0))/(2 π)) + log(z_0) - sum_(k=1)^∞ ((-1)^k (x - z_0)^k z_0^(-k))/k

    log(x) = sum_(j=1)^∞ Res_(s=-j) ((-1 + x)^(-s) Γ(-s)^2 Γ(1 + s))/Γ(1 - s) for abs(-1 + x)<1

    log(x) = Res_(s=0) ((-1 + x)^(-s) Γ(-s) Γ(1 + s))/s + sum_(j=1)^∞ Res_(s=j) ((-1 + x)^(-s) Γ(-s) Γ(1 + s))/s for abs(-1 + x)>1

    Integral representations

    log(x) = integral_1^x 1/t dt

    log(x) = -i/(2 π) integral_(-i ∞ + γ)^(i ∞ + γ) ((-1 + x)^(-s) Γ(-s)^2 Γ(1 + s))/Γ(1 - s) ds for (-1<γ<0 and abs(arg(-1 + x))<π)

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