STUDENTS WILL BE ABLE TO: CONVERT BETWEEN EXPONENT AND LOG FORMS SOLVE LOG EQUATIONS OF FORM LOG B...
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Transcript of STUDENTS WILL BE ABLE TO: CONVERT BETWEEN EXPONENT AND LOG FORMS SOLVE LOG EQUATIONS OF FORM LOG B...
S T U D E N T S W I L L B E A B L E T O :• CONVERT BETWEEN EXPONENT AND LOG FORMS• SOLVE LOG EQUATIONS OF FORM LOGBY=X FOR B, Y, AND X
LOGARITHMIC FUNCTIONS
LOGARITHMS
• For , , and ,if and only if
• The function given by
is called the logarithmic function with base
Read as “log base a
of x”
LOGARITHMS ARE EXPONENTS!
• “What must I raise to in order to get ?• because 2 must be raised to the 3rd power to get 8.
• Why? Logs are a useful way of writing exponents that we don’t know. For example: What must you raise 3 to in order to get 42?
• The log is just a way of writing the exponent!
• Converting between exponential and log form:
COMMON LOGARITHMS
• A log in base 10 is called a common logarithm.
• We can write as
• The LOG button on your calculator evaluates common logs only
• for and
PROPERTIES OF LOGS
1. because
2. because
3. and (they are inverse functions)
4. If , then (one to one)
GRAPHS OF LOGARITHMIC FUNCTIONS
• Graph
• is the inverse function of • To graph: reflect the graph of over the
line (flip the x and y)
PROPERTIES OF LOG FUNCTIONS:
Domain: (0, ∞)Range: (- ∞, ∞)x-intercept: (1, 0)y-int: noneIncreasing: (0, ∞)Vertical Asymptote: x=0HA: noneContinuous: YesReflection of the graph y = ax in the line y=x f(x)=log x
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
y
f(x)=log x
f(x)=10^x
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
y
f(x)=log x
f(x)=10^x
f(x)=x
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
y
STEPS TO GRAPHING LOG FUNCTIONS
• Identify the base of the log and write it as an exponential function
• Make a table of values
• Flip the x and y values to get log function (the inverse)
• Now apply translations:
• Inside parenthesis: change values• Divide by
• Add (remember, if it’s you should subtract 2)
• Outside parenthesis: change values• Multiply by
• Add
TRANSFORMATIONS OF LOG FUNCTIONS
𝑔 (𝑥)=log2 (𝑥−2)
x y
-2
-1
0
1
2
x y x y
2𝑥 log 2𝑥 log 2(𝑥−2)
Domain:
Range:
VA:
x-int:
FIND THE DOMAIN
1. f(x) = log7(x-4)
2. g(x)= log (1-2x)
3. g(x)= log8 x2
How do you think we
can find the domain?
Remember, we can’t take the
log of a negative number.
Set the piece inside the log greater than or equal to zero, then
solve!
• Since we can choose any ‘c’ we want, we’ll choose base 10 (the common log) which the calculator can do
• Ex1: Find
• Ex2: Find
CHANGE OF BASE FORMULA
Any number you want!
THE NATURAL LOGARITHM
• Base • → read “el en of x” or “the
natural log of x”• Inverse of • Definition for Natural Log:
For if and only if
• Use LN button on your calculator to evaluate.
QUIZ 3.1-3.2 THURSDAY
• 3.1 Exponential Functions and their graphs• Evaluate exponential functions• Graph exponential functions• Properties of exp graphs• Natural base• Graph • Evaluate
• Compound Interest• Solve 1-to1 exp equations
• 3.2 Logarithmic Functions and their graphs• Evaluate log functions• Graph log functions• Properties of log graphs• Properties of logs• Common Log• Change of Base• Natural Log • Evaluate