Post on 22-Feb-2016
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Lesson 5-5
Logarithms
Logarithmic functions
Logarithmic functionsThe inverse of
the exponential function.
Logarithmic functionsThe inverse of
the exponential function.
Basic exponential function: f(x) = bx
Logarithmic functionsThe inverse of
the exponential function.
Basic exponential function: f(x) = bx
Logarithmic functionsThe inverse of
the exponential function.
Basic logarithmic function: f-1(x) = logbx
Logarithmic functionsThe inverse of
the exponential function.
Basic logarithmic function: f-1(x) = logbx
Logarithmic functionsThe inverse of
the exponential function.
Basic logarithmic function: f-1(x) = logbx
Every (x,y) (y,x)
Logarithmic functionsBasic rule for changing exponential equations to logarithmic equations
(or vice-versa):
Logarithmic functionsBasic rule for changing exponential equations to logarithmic equations
(or vice-versa):
logbx = a ba = x
Logarithmic functionsBasic rule for changing exponential equations to logarithmic equations
(or vice-versa):
logbx = a ba = x
The base of the logarithmic form becomes the base of the
exponential form.
Logarithmic functionsBasic rule for changing exponential equations to logarithmic equations
(or vice-versa):
logbx = a ba = x
The answer to the log statement becomes the power
in the exponential form.
Logarithmic functionsBasic rule for changing exponential equations to logarithmic equations
(or vice-versa):
logbx = a ba = x
The number you are to take the log of in the log
form, becomes the answer in the exponential form.
Examples:
Examples: log525 = 2 because 52 = 25
Examples: log525 = 2 because 52 = 25
log5125 = 3 because 53 = 125
Examples: log525 = 2 because 52 = 25
log5125 = 3 because 53 = 125
log2(1/8) = - 3 because 2-3 = 1/8
base b exponential function f(x) = bx
base b exponential function f(x) = bx
Domain: All realsRange: All positive reals
base b logarithmic function f-1(x) = logb(x)
base b logarithmic function f-1(x) = logb(x)
Domain: All positive realsRange: All reals
Types of Logarithms
Types of LogarithmsThere are two special logarithms
that your calculator is programmed for:
Types of LogarithmsThere are two special logarithms
that your calculator is programmed for:
log10(x) called the common logarithm
Types of LogarithmsThere are two special logarithms
that your calculator is programmed for:
log10(x) called the common logarithm
For the common logarithm we do not include the
subscript 10, so all you will see is: log (x)
Types of LogarithmsThere are two special logarithms
that your calculator is programmed for:
So, log10(x) log (x) = k if 10k = x
Types of LogarithmsThere are two special logarithms
that your calculator is programmed for:
loge(x) called the natural logarithm
Types of LogarithmsThere are two special logarithms
that your calculator is programmed for:
loge(x) called the natural logarithm
For the natural logarithm, we do not include the subscript
e, so all you will see is: ln (x)
Types of LogarithmsThere are two special logarithms
that your calculator is programmed for:
So, loge(x) ln (x) = k if ek = x
Examples:
Examples:log 6.3 = 0.8 because 100.8 = 6.3
Examples:log 6.3 = 0.8 because 100.8 = 6.3
ln 5 = 1.6 because e1.6 = 5
Example:
Example:Find the value of x to the nearest hundredth.
Example:Find the value of x to the nearest hundredth.
Example:Find the value of x to the nearest hundredth.
10x = 75
Example:Find the value of x to the nearest hundredth.
10x = 75
This transfers to the log statement log 10 75 = x
and the calculator will tell you x = 1.88
Example:Find the value of x to the nearest hundredth.
ex = 75
Example:Find the value of x to the nearest hundredth.
ex = 75
This transfers to the log statement ln 75 = x
and the calculator will tell you x = 4.32
Evaluate:
Evaluate:
Evaluate:
Evaluate:
Evaluate:
Solve:
Solve:
Solve:
Solve:
Assignment:
Pg. 194C.E. #1 – 9 all
W.E. #2 – 14 evens