Definition of logarithmic function Show
: a function (such as y = loga x or y = ln x) that is the inverse of an exponential function (such as y = ax or y = ex) so that the independent variable appears in a logarithm Examples of logarithmic function in a SentenceRecent Examples on the Web Prandtl and von Kármán also discovered that the inertial layer’s mean velocity was a logarithmic function of the distance from the boundary. — Rachel Crowell, Scientific American, 8 Feb. 2022 These example sentences are selected automatically from various online news sources to reflect current usage of the word 'logarithmic function.' Views expressed in the examples do not represent the opinion of Merriam-Webster or its editors. Send us feedback. First Known Use of logarithmic function1836, in the meaning defined above Learn More About logarithmic functionStatistics for logarithmic functionCite this Entry “Logarithmic function.” Merriam-Webster.com Dictionary, Merriam-Webster, https://www.merriam-webster.com/dictionary/logarithmic%20function. Accessed 25 Oct. 2022. Love words? Need even more definitions? Subscribe to America's largest dictionary and get thousands more definitions and advanced search—ad free! Merriam-Webster unabridged
In Mathematics, before the discovery of calculus, many Math scholars used logarithms to change multiplication and division problems into addition and subtraction problems. In Logarithms, the power is raised to some numbers (usually, base number) to get some other number. It is an inverse function of exponential function. We know that Mathematics and Science constantly deal with the large powers of numbers, logarithms are most important and useful. In this article, we are going to discuss the definition and formula for the logarithmic function, rules and properties, examples in detail. Logarithmic Function DefinitionIn mathematics, the logarithmic function is an inverse function to exponentiation. The logarithmic function is defined as For x > 0 , a > 0, and a \(\begin{array}{l}\neq\end{array} \) 1,y= loga x if and only if x = ay Then the function is given by f(x) = loga x The base of the logarithm is a. This can be read it as log base a of x. The most 2 common bases used in logarithmic functions are base 10 and base e. Also, try out: Logarithm Calculator Common Logarithmic FunctionThe logarithmic function with base 10 is called the common logarithmic function and it is denoted by log10 or simply log. f(x) = log10 x Natural Logarithmic FunctionThe logarithmic function to the base e is called the natural logarithmic function and it is denoted by loge. f(x) = loge x Logarithmic Functions have some of the properties that allow you to simplify the logarithms when the input is in the form of product, quotient or the value taken to the power. Some of the properties are listed below. Product Rule logb MN = logb M + logb N Multiply two numbers with the same base, then add the exponents. Example : log 30 + log 2 = log 60 logb M/N = logb M – logb N Divide two numbers with the same base, subtract the exponents. Example : log8 56 – log8 7 = log8(56/7)=log88 = 1 Power Rule Raise an exponential expression to power and multiply the exponents. Logb Mp = P logb M Example : log 1003 = 3. Log 100 = 3 x 2 = 6 Zero Exponent Rule loga 1 = 0. Change of Base Rule logb (x) = ln x / ln b or logb (x) = log10 x / log10 b Other Important Rules of Logarithmic Function
There are also some of the logarithmic function with fractions. It has a useful property to find the log of a fraction by applying the identities
We also can have logarithmic function with fractional base. Consider an example, \(\begin{array}{l}3\log _{\frac{4}{9}}\sqrt[4]{\frac{27}{8}}=\frac{3}{4}\log _{\frac{4}{9}}\frac{27}{8}\end{array} \) By the definition, loga b = y becomes ay = b (4/9)y = 27/8 (22/32)y = 33 / 23 (⅔)2y = (3/2)3 Video LessonLogarithmic EquationsLogarithmic Function ExamplesHere you are provided with some logarithmic functions example. Example 1: Use the properties of logarithms to write as a single logarithm for the given equation: 5 log9 x + 7 log9 y – 3 log9 z Solution: By using the power rule , Logb Mp = P logb M, we can write the given equation as 5 log9 x + 7 log9 y – 3 log9 z = log9 x5 + log9 y7 – log9 z3 From product rule, logb MN = logb M + logb N 5 log9 x + 7 log9 y – 3 log9 z = log9 x5y7 – log9 z3 From Quotient rule, logb M/N = logb M – logb N 5 log9 x + 7 log9 y – 3 log9 z = log9 (x5y7 / z3 ) Therefore, the single logarithm is 5 log9 x + 7 log9 y – 3 log9 z = log9 (x5y7 / z3 ) Question 2: Use the properties of logarithms to write as a single logarithm for the given equation: 1/2 log2 x – 8 log2 y – 5 log2 z Solution: By using the power rule , Logb Mp = P logb M, we can write the given equation as 1/2 log2 x – 8 log2 y – 5 log2 z = log2 x1/2 – log2 y8 – log2 z5 From product rule, logb MN = logb M + logb N Take minus ‘- ‘ as common 1/2 log2 x – 8 log2 y – 5 log2 z = log2 x1/2 – log2 y8z5 From Quotient rule, logb M/N = logb M – logb N 1/2 log2 x – 8 log2 y – 5 log2 z = log2 (x1/2 / y8z5 ) The solution is 1/2 log2 x – 8 log2 y – 5 log2 z = \(\begin{array}{l}\log _{2}\left ( \frac{\sqrt{x}}{y^{8}z^{5}} \right )\end{array} \) For more related articles on logarithmic function and its properties, register with BYJU’S – The Learning app and watch interactive videos. What is logarithmic function and example?A logarithm is an exponent. Any exponential expression can be rewritten in logarithmic form. For example, if we have 8 = 23, then the base is 2, the exponent is 3, and the result is 8. This can be rewritten in logarithmic form as. 3 = log2 8.
What is a logarithmic function in simple terms?Definition of logarithmic function
: a function (such as y = loga x or y = ln x) that is the inverse of an exponential function (such as y = ax or y = ex) so that the independent variable appears in a logarithm.
What are logarithmic functions?A logarithm is a mathematical operation that determines how many times a certain number, called the base, is multiplied by itself to reach another number.
How do you determine if it is logarithmic function?Definition of Logarithm
logb x = y if and only if by = x, where x > 0 and b > 0, b ≠ 1.
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