Logarithmic function
Done By:
Al-Hanoof
Amna
Dana
Ghada
Logarithmic Function
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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 oneto-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
ay=Х
Clearly, Loga Х is the exponent to which the base a must be
raised to give Х
Logarithmic Function
Logarithmic form
Exponent
Loga x = y
Base
Exponential form
Exponent
a^ y = Х
Base
Logarithmic Function
Example:
Logarithmic form
Log2 8= 3
Exponential form
2^3=8
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.
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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
y = loga x
y
4
3
2
1
x
-4
-3
-2
-1
1
2
3
4
-1
-2
-3
-4
This is the basic function y= Loga x
Logarithmic Function
y =- loga x
y
4
3
2
1
x
-4
-3
-2
-1
1
2
3
4
-1
-2
-3
-4
The function is reflected in the x-axis.
Logarithmic Function
y = log2 (-x)
y
4
3
2
1
x
-4
-3
-2
-1
1
2
3
4
-1
-2
-3
-4
The function is reflected in the y-axis.
Logarithmic Function
Y=loga(x+2)
y
4
3
2
1
x
-4
-3
-2
-1
1
2
3
4
-1
-2
-3
-4
The function is shifted to the left by two unites .
Logarithmic Function
y = loga (x-2)
y
4
3
2
1
x
-4
-3
-2
-1
1
2
3
4
-1
-2
-3
-4
The function is shifted to the right by two unites .
Logarithmic Function
y = logax +2
y
4
3
2
1
x
-4
-3
-2
-1
1
2
3
4
-1
-2
-3
-4
The function is shifted to the upward by two unites .
Logarithmic Function
y = loga x -2
y
4
3
2
1
x
-4
-3
-2
-1
1
2
3
4
-1
-2
-3
-4
The function is shifted to the downward by two unites .
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
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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
y
4
Y=e^x
3
2
1
x
-4
Y=ln x
-3
-2
-1
1
-1
-2
-3
-4
2
3
4
Logarithmic Function
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
*WARNING:
loga (x+y) ≠ logax +logay
Log 6/log2 ≠ log(6/2)
(log2x)3 ≠ 3log2x
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
y log a b =logax
y=(loga x)/(loga b)
(take loga for both sides)
(law3)
(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 3
x = log7 -2
(subtract by 2)
log3
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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 year
t 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
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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) =
$10,000
r = 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
0.09t = In 2
t=(In 2)/(0.09)
t ≈7.702
The money will double in 7.7 years.
(Take 10 of each side)
(Property of In)
(Divide by 0.09)
(Use a calculator)