LOGARITHMIC FUNCTIONS Presented by: AMEENA AMEEN MARYAM BAQIR FATIMA EL MANNAI KHOLOOD REEM IBRAHIM MARIAM OSAMA. Content:. Definition of logarithm How to write a Logarithmic form as an Exponantional form Properties of logarithm Laws of logarithm Changing the base of log Common logarithm.
The power to which a base must be raised to yield a given number
the logarithm to the base 3 of 9, or log3 9, is 2, because 32 = 9
Logarithm of products
Logarithm of quotient
Logarithm of a power
The log of 1 is always equal to 0 but the log of a number which is similar to the base of log is always equal 1
Transform the addition
Transforming the subtraction
Will be changed into
And the same for
Let a, b, and x be positive real numbers such that and (remember x must be greater than 0). Then can be converted to the base b by the formula
Take the base-c logarithm of each side
Divide each side by
If a and b are positive numbers not equal to 1 and M is positive,
If the new base is 10 or e, then:
the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm,[ ] .
The binary logarithm is the logarithm for base 2. It is the inverse function of .
the binary logarithm is the logarithm for base 2. It is the inverse function of .
Is the logarithm to the base e, where e is an irrational constant approximately equal to 2.718281828459.
The natural logarithm can be defined for all positive real numbers x as the area under the curve y = 1/t from 1 to x, and can also be defined for non-zero complex numbers.
The natural logarithm function can also be defined as the inverse function of the exponential function, leading to the identities:
Logarithmic equations contain logarithmic expressions and constants. When one side
of the equation contains a single logarithm and the other side contains a constant, the
equation can be solved by rewriting the equation as an equivalent exponential equation using the definition of logarithm.
The natural log of x does not equal 1/x, however the derivative of ln(x) does:
The derivative of log(x) is given as:d/dx[ log-a(x) ] = 1 / (x * ln(a))where "log-a" is the logarithm of base a.
However, when a = e (natural exponent),
then log-a(x) becomes ln(x) and ln(e) = 1:
d/dx[ log-e(x) ] = 1 / (x * ln(e))
d/dx[ ln(x) ] = 1 / (x * ln(e))
d/dx[ ln(x) ] = 1 / (x * 1)
d/dx[ ln(x) ] = 1 / x
Before graphing y= logb (x) we can start first with knowing the following:
The logarithm of 1 is zero (x=1), so the x-intercept is always 1, no matter what base of log was.
For example if we have:
b = 2 power 0 = 1
b = 3 power 0 = 1
b = 4 power 0 = 1
Values of x between 0 and 1 represent the graph below the x-axis when:
Fractions are the values of the negative powers.
Graph y = log2(x).
First change log to exponent form:
X=2 power y, then start with a T-chart
First change ln into logarithm form:
Then change to exponential form:
X= e power y..Now draw you T-chart
Graph y = log2(x + 3).
This is similar to the graph of log2(x),
but is shifted
"+ 3" is not outside of the log,
the shift is not up or down
First plot (1,0), test the shifting
The log will be 0 when the argument,
x + 3, is equal to 1.
When x = –2. (1, 0) the basic point is shifted to
So, the graph is shifted three units to the left
draw the asymptote on the x= -3
y = loga x
y = log2 (-x)
y = logax +2