Logarithmic function Done By: Al-Hanoof Amna Dana Ghada
Dec 19, 2015
Logarithmic Function
Changing from Exponential to Logarithmic form. ( Ghada )
Graphing of Logarithmic Function.( Amna ) Common Logarithm.( Ghada ) Natural Logarithm.( Ghada ) Laws of Logarithmic Function. ( Dana ) Change of the base.( Ghada ) Solving of Logarithmic Function.( Al-Hanoof ) Application of Logarithmic Function.(Al-Hanoof)
Logarithmic Function
Every exponential function f(x) = a x, with a > 0 and a ≠ 1. is a one-to-one function, therefore has an inverse function(f-1). The inverse function is called the Logarithmic function with base a and is denoted by Loga
Let a be a positive number with a ≠ 1. The logarithmic function with base a, denoted by loga is defined by:
Loga x = y a y = Х Clearly, Loga Х is the exponent to which the base a must be raised to give Х
Logarithmic Function
Logarithmic form Exponential form
Exponent Exponent
Loga x = y a^ y = Х
Base Base
Logarithmic Function
Graphs of Logarithmic Functions:
The exponential function f(x) =a^x has
Domain: IR
Range: (0.∞),
Since the logarithmic function is the inverse function for the exponential function , it has
Domain : (0, ∞)
Range: IR.
Logarithmic Function
The graph of f(x) = Loga x is obtained by reflecting the graph of f(x) = a^ x the line y = x
x-intercept of the function y = Loga x is 1
f(x) = a^ x
y = x
Logarithmic Function
This is the basic function y= Loga x
y = loga x
1 2 3 4-1-2-3-4
x1
2
3
4
-1
-2
-3
-4
y
Logarithmic Function
y =- loga x
The function is reflected in the x-axis .
1 2 3 4-1-2-3-4
x1
2
3
4
-1
-2
-3
-4
y
Logarithmic Function
y = log2 (-x)
The function is reflected in the y-axis .
1 2 3 4-1-2-3-4
x1
2
3
4
-1
-2
-3
-4
y
Logarithmic Function
The function is shifted to the left by two unites .
Y=loga(x+2)
1 2 3 4-1-2-3-4
x1
2
3
4
-1
-2
-3
-4
y
Logarithmic Function
The function is shifted to the right by two unites .
y = loga (x-2)
1 2 3 4-1-2-3-4
x1
2
3
4
-1
-2
-3
-4
y
Logarithmic Function
The function is shifted to the upward by two unites .
y = logax +2
1 2 3 4-1-2-3-4
x1
2
3
4
-1
-2
-3
-4
y
Logarithmic Function
y = loga x -2
The function is shifted to the downward by two unites .
1 2 3 4-1-2-3-4
x1
2
3
4
-1
-2
-3
-4
y
Logarithmic Function
Example:Finding the domain of a logarithmic function: F(x)=log(x-2)Solution:As any logarithmic function lnx is defined when x>0,
thus, the domain of f(x) is x-2 >0 X>2 So the domain =(2,∞)
Logarithmic Function
Common Logarithmic;
The logarithm with base 10 is called the common logarithm and is denoted by omitting the base:
log x = log10x
Natural Logarithms:
The logarithm with base e is called the natural logarithm and is denoted by In:
ln x =logex
Logarithmic Function
The natural logarithmic function y = In x is the inverse function of the exponential function y = e^X.By the definition of inverse functions we have:
ln x =y e^y=x
1 2 3 4-1-2-3-4
x1
2
3
4
-1
-2
-3
-4
y
Y=e^x
Y=ln x
Laws of logarithms:
Let a be a positive number, with a≠1. let A>0, B>0, and C be any real numbers.
1. loga (AB) = loga A + loga B
log2 (6x) = log2 6 + log2 x
2. loga (A/B) = loga A - loga B
log2 (10/3) = log2 10 – log2 3
3. loga A^c = C loga A
log3 √5 = log3 51/2 = 1/2 log3 5
Logarithmic Function
Rewrite each expression using logarithm laws
log5 (x^3 y^6)
= log5 x^3 + log5 y^6 law1
= 3 log5 x + 6 log5 y law3
ln (ab/3√c)
= ln (ab) – ln 3√c law2
= ln a + ln b – ln c1/3 law1
= ln a + ln b – 1/3 ln c law3
Logarithmic Function
Express as a single logarithm
3 log x + ½ log (x+1)
= log x^3 + log (x+1)^1/2 law3
=log x^3(x+1)^1/2 law1
3 ln s + ½ ln t – 4 ln (t2+1)
= ln s^3 + ln t^1/2 – ln (t^2+1)^4 law3
= ln ( s^3 t^1/2) – ln (t^2 + 1)^4 law1
= ln s^3 √t /(t2+1)^4 law2
Logarithmic Function
Logarithmic Function
Change of Base:Sometimes we need to change from logarithms in one
base to logarithms in another base. b^y = x (exponential form)
logab^y = logax (take loga for both sides)
y log a b =logax (law3)
y=(loga x)/(loga b) (divide by logab)
Logarithmic Function
Example:
Since all calculators are operational for log10 we will change the base to 10
Log8 5 = log10 5/ log10 8≈ 0.77398 (approximating the answer by using the calculator)
Logarithmic Function
Solving the logarithmic Equations:Example:Find the solution of the equation log 3^(x+2) = log7.SOLUTION:
) x + 2 (log 3=log7 (bring down the exponent)X+2= log7 (divide by log 3 ) log 3x = log7 -2 (subtract by 2) log3
Logarithmic Function
Application of e and Exponential Functions:In the calculation of interest exponential function is used. In order to
make the solution easier we use the logarithmic function.A= P (1+ r/n)^nt
A is the money accumulated.P is the principal (beginning) amount r is the annual interest rate n is the number of compounding periods per yeart is the number of years
There are three formulas:
A = p(1+r) Simple interest (for one year)
A(t) = p(1+r/n)nt Interest compounded n times per year
A(t) = pert Interest compounded continuously
Logarithmic Function
Example: A sum of $500 is invested at an interest rate 9%per year. Find the time required for the money to double if the interest is compounded according to the following method.
a) Semiannual b) continuous Solution:
)a( We use the formula for compound interest with P = $5000, A (t) = $10,000r = 0.09, n = 2, and solve the resulting exponential equation for t.
(1.045)^2t = 2 (Divide by 5000)log (1.04521)^2t = log 2 (Take log of each side)2t log 1.045 = log 2 Law 3 (bring down the exponent)t= (log 2)/ (2 log 1.045) (Divide by 2 log 1.045)t ≈ 7.9 The money will double in 7.9 years. (using a calculator)
Logarithmic Function
(b) We use the formula for continuously compounded interest with P = $5000,
A(t) = $10,000, r = 0.09, and solve the resulting exponential equation
for t.
5000e^0.09t = 10,000
e^0.091 = 2 (Divide by 5000)
In e0.091 = In 2 (Take 10 of each side)
0.09t = In 2 (Property of In)
t=(In 2)/(0.09) (Divide by 0.09)
t ≈7.702 (Use a calculator)
The money will double in 7.7 years.