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## Logarithm

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**Logarithm (Introduction)**The logarithmic function is defined as the inverse of the exponential function. *A LOGARITHM is an exponent. It is the exponent to which the base must be raised to produce a given number. For b > 0 and b 1 is equivalent to Number Exponent Base**Examples,**• Since , then • Since , then • Since , then • Since , then**Properties of Logarithm**• because • because • because**Rules of Logarithm**1. 2. 3.**Example:-**1)**Changing the base:**IF: ,Then Y= Now we solve For y, using base-b logarithms: If: Take the base-b logarithm of each side Power rule Divide each side by**Base-change formula:**If a and b are positive numbers not equal to 1 and M is positive, then**Example :find To four decimal**places • Solution: By using the base-change with a=7 and b=10: Chick by finding with calculator. Note that we also have**Common logarithms**• common logarithm is the logarithm with base 10. • It is indicated by or sometimes Log(x) with a capital L • Traditionally, log10 is abbreviated to log.**Binary logarithm**• In mathematics, the binary logarithm is the logarithm for base 2. It is the inverse function of . • Domain and range: the domain of the exponential function is and its range is Because the logarithm function is the inverse of The domain of is and its range is**Logarithmic equation :**If we have equality of two logarithms with the same base, we use the one-to-one property to eliminate the logarithm. If we have an equation with only one logarithm, such we use the definition of logarithm to write and to eliminate the logarithm**Find the solution :**• Solution: Original equation Take log of each side Power rule Distributive property Get all x-terms on one side Factor out x Exact solution**Example (2)**Solve Solution : Original equation Product rule Multiply the binomials Definition of logarithm Even root property**To check, first let x=-5 in the original equation :**Because the domain of any logarithmic function is the set of positive numbers, these logarithms are undefined. Now check x=5 in the originalequation: The solution is {5}. Incorrect Correct**Natural Logarithm**• The natural logarithm is a logarithm to base e • Where e = 2.7182818…. • it is denoted ln x, as ln x = loge x**Reason for being "natural"**The reason we call the ln(x) "natural" : • expressions in which the unknown variable appears as the exponent of e occur much more often than exponents of 10 • the natural logarithm can be defined quite easily using a simple integral or Taylor series--which is not true of other logarithms • there are a number of simple series involving the natural logarithm, and it often arises in nature. Nicholas Mercator first described them as log naturalis before calculus was even conceived.**The general definition of a logarithm**Y = ln x means the same as x = ey And this leads us directly to the following: • ln 1 = 0 because e0= 1 • ln e = 1 because e1= e • ln e2= 2 and ln e-3= -3**Properties:**• All the usual properties of logarithms hold for the natural logarithm, for example: • (where 28 is an arbitrary real number) • ln (x)a = a ln x**Example 1:**• ln(e)4= 4ln(e) = 4(1) & since ln e = 1**Example 3:**• ln 5e= = ln 5 + ln e = ln ( 5 )+ 1**Example 4:**• It doesn't exist! Why?**Example 5**• e ln 6 = ? • e ln 6 = 6**Proof that d/dx ln(x) = 1/x**• F (x) = ln(x) • . f ‘ (x) = lim h-->0 (f (x + h) – f (x)) /h • Definition of a derivative • = lim h-->0 (ln(x + h) - ln(x))/h • Plugging the function f (x) = ln(x) • = lim h-->0 ln( (x + h) /x) /h • Rule of logarithms: log (a) – log (b) = log (a/b)**= lim h-->0 ln(1 + h/x)/h**• Algebraic simplification: (x + h)/x = 1 + x/h • = lim h-->0 ln(1 + h/x)⋅(x/h)⋅(1/x) • Algebraically, 1/h = (x/h)(1/x) • = 1/x ⋅ lim h-->0 ln(1 + h/x) ⋅ (x/h) • 1/x is a constant with respect to the variable being "limited," so we can pull it out of the limit .**= 1/x ⋅ lim_h-->0 ln((1 + h/x)x/h)**• Rule of logs: log(a) ⋅ b = log(ab) • Let's look at a definition of e using a limit: • e = lim n-->∞ (1 + 1/n)n Or equivalently: e = lim n-->0 (1 + n)1/n • lim h-->0 (1 + h/x) x/h = e • True from the definition of e (the x is irrelevant, since it's constant with respect to h) • 1/x ⋅ lim_h-->0 ln((1 + h/x)x/h)**= 1/x ⋅ ln(e)**• Follows from (7.5) applied to (7)Since e is the base of ln: ln (e) = 1 • = 1/x • What happens when you multiply anything by 1 is that it doesn't change.**(b,1)**1 1 b • for any base ,x-intercept is 1. • because the logarithm of 1 is 0 . 2)The graph passes through the point (b,1) . because the logarithm of the base is 1. 3) The graph is below the x-axis, the logarithm is negative for Which number are those that have negative logarithms.**Ex:(4)Graph the following**Sol: Change to exponential form,**Ex:(5)Graph the following**• Sol: • Change to exponential form,**Sol:**Change to exponential form, Ex:(3)Graph the following.**EX:**Solve S0l: No solution