Download ELF 3

Algebra II with Trigonometry
ELF-3
LOGARITHMIC FUNCTIONS (TB 7-3)
Warm-Up:
Solve 3x = 7
Find the inverse of each of the following functions. Is the inverse a function?
1)
y = 5x – 1
inverse _____________________
function? ___________________
2)
y = x3
inverse _____________________
function? ___________________
Find the inverse of the exponential function y = ax. _________________
The logarithmic function is defined to express the inverse of an exponential
function.
For all positive real numbers x and a, where a ≠ 1,
The inverse of the exponential function y = ax is the logarithmic
function y = loga x
Therefore, y = loga x if and only if x = ay.
We have discussed previously that the irrational number e is a cool number to work with.
The inverse of f(x) = ex is called the natural logarithmic function, which is written ln x
(“the natural log of n”). Since it is written without a base, it is understood to be e.
The Natural Logarithmic Function
The function defined by
f (x) = loge x = ln x,
x>0
is the natural logarithmic function.
RS/CR/MD/TH 2/15
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Algebra II with Trigonometry
Express in logarithmic form.
1)
24 = 16 ________________
2)
163/4 = 8 ________________
3)
27-2/3 = 1/9 ________________
4)
103/2 = 10 10 ________________
5)
34/3 = 3 3 3 ________________
6)
e5 = 148.413
log10 100 = 2 ________________
Express in exponential form.
7)
log2 8 = 3 ________________
8)
9)
log10 0.001 = -3 ________________
10) log3 81 = 4 ________________
11) log5
1
= -3 ________________
125
12) ln 1.6487 = ½
Special Logarithmic Values

loga 1 = 0
because _________

loga a = 1
because _________

loga ax = x
because _________
To evaluate a logarithm, set it equal to a variable and rewrite the equation in exponential
form. Evaluate each of the following.
1
________________
16
11) log6 36 ________________
12) log2
13) log3 3 3 ________________
14) log2 8 2 ________________
RS/CR/MD/TH 2/15
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Algebra II with Trigonometry
Some equations involving logarithms can be solved easily if they are first written in
exponential form.
15) logx 125 = 3
17)
log3 81 = x
x = ______
x = ______
16) log2 x = 4
18) log6
1
=x
216
x = ______
x = ______
The Common Logarithmic Function is the logarithmic function with base 10. This
function can be evaluated on most scientific calculators with the log key.
To use a calculator to evaluate logarithms with other bases, use the change of base
formula.
log10 x
Change of Base Formula loga x =
log10 a
Use a calculator and the change of base formula to evaluate to the ten-thousandths
place.
19) log2 6 ≈ _________
20) log2 13 ≈ _________
21) log6 -12 ≈ _________
22) log1/2 6 ≈ _________
Going back to the warm-up. How would you use logarithms to solve 3x = 7?
RS/CR/MD/TH 2/15
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Algebra II with Trigonometry
ELF-3
LOGARITHMIC FUNCTIONS
I.
Evaluate each of the following logarithms.
1)
log 1000
_________________
2)
log 36 6
_________________
3)
log 4 64
_________________
4)
log 12 0
_________________
5)
log 7 49
_________________
6)
log 9 -81
_________________
7)
log 5
8)
log 16 1611
_________________
9)
log 15 15
_________________
10) log 2 16
_________________
11) log 28 1
_________________
1
25
_________________
II. Rewrite each of the following expressions in exponential form.
1)
log 2 16
_________________
2)
log 1000
_________________
3)
log 28 1
_________________
RS/CR/MD/TH 2/15
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Algebra II with Trigonometry
III. Solve each of the following equations.
1)
log x 125 = 3
_________________
2)
log 12 x = 2
_________________
3)
log 7 343 = x
_________________
RS/CR/MD/TH 2/15
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