Logarithm properties

[ "article:topic", "quotient rule", "power rule", "product rule", "change-of-base formula", "authorname:openstax", "product rule of logarithms", "quotient rule of logarithms", "power rule for logarithms", "calcplot:yes", "license:ccby", "showtoc:yes", "source[1]-math-31105", "source-math-31105", "licenseversion:40", "source@https://openstax.org/details/books/precalculus" ] \( \newcommand\) • • • • • • • • • Skills to Develop • Use the product rule for logarithms. • Use the quotient rule for logarithms. • Use the power rule for logarithms. • Expand logarithmic expressions. • Condense logarithmic expressions. • Use the change-of-base formula for logarithms. In chemistry, • Battery acid: \(0.8\) • Stomach acid: \(2.7\) • Orange juice: \(3.3\) • Pure water: \(7\) at \(25^\circ C\) • Human blood: \(7.35\) • Fresh coconut: \(7.8\) • Sodium hydroxide (lye): \(14\) To determine whether a solution is acidic or alkaline, we find its pH, which is a measure of the number of active positive hydrogen ions in the solution. The pH is defined by the following formula, where \([H^+]\) is the concentration of hydrogen ion in the solution \[\begin_3(2x+5)\) for \(x\). Since the bases are the same, we can apply the one-to-one property by setting the arguments equal and solving for \(x\): \(3x=2x+5\) Set the arguments equal. \(x=5\) Subtract \(2x\). Using the Product Rule for Logarithms What about the equation \(\). We have a similar property for logarithms, called the product rule for logarith...

Product rule log ⁡ b ( M N ) = log ⁡ b ( M ) + log ⁡ b ( N ) \large\log_b(MN)=\log_b(M)+\log_b(N) lo g b ​ ( M N ) = lo g b ​ ( M ) + lo g b ​ ( N ) log, start base, b, end base, left parenthesis, M, N, right parenthesis, equals, log, start base, b, end base, left parenthesis, M, right parenthesis, plus, log, start base, b, end base, left parenthesis, N, right parenthesis Quotient rule log ⁡ b ( M N ) = log ⁡ b ( M ) − log ⁡ b ( N ) \large\log_b\left(\frac lo g b ​ ( M ) = lo g a ​ ( b ) lo g a ​ ( M ) ​ log, start base, b, end base, left parenthesis, M, right parenthesis, equals, start fraction, log, start base, a, end base, left parenthesis, M, right parenthesis, divided by, log, start base, a, end base, left parenthesis, b, right parenthesis, end fraction For example, we can use the product rule to rewrite log ⁡ ( 2 x ) \log(2x) lo g ( 2 x ) log, left parenthesis, 2, x, right parenthesis as log ⁡ ( 2 ) + log ⁡ ( x ) \log(2)+\log(x) lo g ( 2 ) + lo g ( x ) log, left parenthesis, 2, right parenthesis, plus, log, left parenthesis, x, right parenthesis . Because the resulting expression is longer, we call this an expansion. In another example, we can use the change of base rule to rewrite ln ⁡ ( x ) ln ⁡ ( 2 ) \dfrac ln ( 2 ) ln ( x ) ​ start fraction, natural log, left parenthesis, x, right parenthesis, divided by, natural log, left parenthesis, 2, right parenthesis, end fraction as log ⁡ 2 ( x ) \log_2(x) lo g 2 ​ ( x ) log, start base, 2, end base, left parenthesis, x, r...

Properties of Logarithms In Mathematics, properties of logarithms functions are used to solve logarithm problems. We have learned many properties in basic maths such as commutative, associative and distributive, which are applicable for algebra. In the case of Table of Contents: • • • • • • • • • • The logarithmic number is associated with exponent and power, such that if x n = m, then it is equal to log x m=n. Hence, it is necessary that we should also learn 4 = 10000, so log 1010000 = 4. With the help of these properties, we can express the logarithm of a product as a sum of logarithms, the log of the quotient as a difference of log and log of power as a product. Only positive real numbers have real number logarithms, negative and complex numbers have complex logarithms. Logarithm Base Properties Before we proceed ahead for logarithm properties, we need to revise the law of exponents, so that we can compare the properties. For exponents, the laws are: • Product rule: a m.a n=a m+n • Quotient rule: a m/a n = a m-n • Power of a Power: (a m) n = a mn Now let us learn the properties of logarithmic functions. Product Property If a, m and n are positive integers and a ≠1, then; log a(mn) = log am + log an Thus, the log of two numbers m and n, with base ‘a’ is equal to the sum of log m and log n with the same base ‘a’. Example: log 3(9.25) = log 3(9) + log 3(27) = log 3(3 2) + log 3(3 3) = 2 + 3 (By property: log b b x = x) = 5 Quotient Property If m, n and ...

Natural Logarithm - ln(x) Natural logarithm is the logarithm to the base e of a number. • • • • • • • • When e y = x Then base e logarithm of x is ln( x) = log e( x) = y The e≈ 2.71828183 Ln as inverse function of exponential function The natural logarithm function ln(x) is the inverse function of the exponential function e x. For x>0, f ( f -1( x)) = e ln( x) = x Or f -1( f ( x)) = ln( e x) = x Rule name Rule Example ln( x ∙ y) = ln( x) + ln( y) ln(3 ∙7) = ln(3) + ln(7) ln( x / y) = ln( x) - ln( y) ln(3 / 7) = ln(3) - ln(7) ln( x y) = y ∙ln( x) ln(2 8) = 8 ∙ln(2) f ( x) = ln( x) ⇒ f ' ( x) = 1 / x ∫ ln( x) dx = x ∙(ln( x) - 1) + C ln of negative number ln( x) is undefined when x ≤ 0 ln(0) is undefined ln(1) = 0 lim ln( x) = ∞ , when x→∞ Euler's identity ln(-1) = iπ The logarithm of the multiplication of x and y is the sum of logarithm of x and logarithm of y. log b( x ∙ y) = log b( x) + log b( y) For example: log 10(3 ∙7) = log 10(3) + log 10(7) The logarithm of the division of x and y is the difference of logarithm of x and logarithm of y. log b( x / y) = log b( x) - log b( y) For example: log 10(3 / 7) = log 10(3) - log 10(7) The logarithm of x raised to the power of y is y times the logarithm of x. log b( x y) = y ∙log b( x) For example: log 10(2 8) = 8 ∙log 10(2) The derivative of the natural logarithm function is the reciprocal function. When f ( x) = ln( x) The derivative of f(x) is: f ' ( x) = 1 / x The integral of the natural logarithm function is given by: When f...

Product Rule: log ⁡ b ( M N ) = log ⁡ b ( M ) + log ⁡ b ( N ) \log_b(MN)=\log_b(M)+\log_b(N) lo g b ​ ( M N ) = lo g b ​ ( M ) + lo g b ​ ( N ) log, start base, b, end base, left parenthesis, M, N, right parenthesis, equals, log, start base, b, end base, left parenthesis, M, right parenthesis, plus, log, start base, b, end base, left parenthesis, N, right parenthesis log ⁡ 2 ( 4 ⋅ 8 ) = log ⁡ 2 ( 2 2 ⋅ 2 3 ) 2 2 = 4 and 2 3 = 8 = log ⁡ 2 ( 2 2 + 3 ) a m ⋅ a n = a m + n = 2 + 3 log ⁡ b ( b c ) = c = log ⁡ 2 ( 4 ) + log ⁡ 2 ( 8 ) Since 2 = log ⁡ 2 ( 4 ) and 3 = log ⁡ 2 ( 8 ) \begin lo g 2 ​ ( 4 ⋅ 8 ) ​ = lo g 2 ​ ( 2 2 ⋅ 2 3 ) = lo g 2 ​ ( 2 2 + 3 ) = 2 + 3 = lo g 2 ​ ( 4 ) + lo g 2 ​ ( 8 ) ​ ​ 2 2 = 4 and 2 3 = 8 a m ⋅ a n = a m + n l o g b ​ ( b c ) = c Since 2 = l o g 2 ​ ( 4 ) and 3 = l o g 2 ​ ( 8 ) ​ And so we have that log ⁡ 2 ( 4 ⋅ 8 ) = log ⁡ 2 ( 4 ) + log ⁡ 2 ( 8 ) \log_2()=\log_2(4)+\log_2(8) lo g 2 ​ ( 4 ⋅ 8 ) = lo g 2 ​ ( 4 ) + lo g 2 ​ ( 8 ) log, start base, 2, end base, left parenthesis, 4, dot, 8, right parenthesis, equals, log, start base, 2, end base, left parenthesis, 4, right parenthesis, plus, log, start base, 2, end base, left parenthesis, 8, right parenthesis . Notice, that writing 4 4 4 4 and 8 8 8 8 as powers of 2 2 2 2 was key to the proof. So in general, we'd like M M M M and N N N N to be powers of the base b b b b . To do this, we can let M = b x M=b^x M = b x M, equals, b, start superscript, x, end superscript and N = b y N=b^y N = b y N, equals...

Logarithms In Mathematics, logarithms are the other way of writing the exponents. A logarithm of a number with a base is equal to another number. A logarithm is just the opposite function of exponentiation. For example, if 10 2 = 100 then log 10 100 = 2. Hence, we can conclude that, Log b x = n or  b n = x Where b is the base of the logarithmic function. This can be read as “Logarithm of x to the base b is equal to n”. In this article, we are going to learn the definition of logarithms, two types of logarithms such as common logarithm and natural logarithm, and different properties of logarithms with many solved examples. Table of contents: • • • • • • • History John Napier introduced the concept of Logarithms in the 17th century. Later it was used by many scientists, navigators, engineers, etc for performing various calculations which made it simple. In simple words, Logarithms are the inverse process of exponentiation. What are Logarithms? A logarithm is defined as the power to which a number must be raised to get some other values. It is the most convenient way to express “The logarithm of a positive real number a with respect to base b, a positive real number not equal to 1 [nb 1], is the exponent by which b must be raised to yield a”. i.e. b y= a ⇔ log b a =y Where, • “a” and “b” are two positive real numbers • y is a real number • “a” is called argument, which is inside the log • “b” is called the base, which is at the bottom of the log. In other words, the...

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3 Polynomial and Rational Functions • Introduction to Polynomial and Rational Functions • 3.1 Complex Numbers • 3.2 Quadratic Functions • 3.3 Power Functions and Polynomial Functions • 3.4 Graphs of Polynomial Functions • 3.5 Dividing Polynomials • 3.6 Zeros of Polynomial Functions • 3.7 Rational Functions • 3.8 Inverses and Radical Functions • 3.9 Modeling Using Variation • 4 Exponential and Logarithmic Functions • Introduction to Exponential and Logarithmic Functions • 4.1 Exponential Functions • 4.2 Graphs of Exponential Functions • 4.3 Logarithmic Functions • 4.4 Graphs of Logarithmic Functions • 4.5 Logarithmic Properties • 4.6 Exponential and Logarithmic Equations • 4.7 Exponential and Logarithmic Models • 4.8 Fitting Exponential Models to Data • 7 Trigonometric Identities and Equations • Introduction to Trigonometric Identities and Equations • 7.1 Solving Trigonometric Equations with Identities • 7.2 Sum and Difference Identities • 7.3 Double-Angle, Half-Angle, and Reduction Formulas • 7.4 Sum-to-Product and Product-to-Sum Formulas • 7.5 Solving Trigonometric Equations • 7.6 Modeling with Trigonometric Functions • 8 Further Applications of Trigonometry • Introduction to Further Applications of Trigonometry • 8.1 Non-right Triangles: Law of Sines • 8.2 Non-right Triangles: Law of Cosines • 8.3 Polar Coordinates • 8.4 Polar Coordinates: Graphs • 8.5 Polar Form of Complex Numbers • 8.6 Parametric Equations • 8.7 Parametric Equations: Graphs • 8.8 Vectors • 9 Systems of...