REVIEW 4
1) If the domain of a one-to-one function f is (-2,), the range of its inverse is (-2,)
2) The domain of the logarithmic function f(x) = logax is {x|x>0 or (0,)
3) log2√2
½
4) If Mark has $100 to invest at 8% per annum compounded semiannually, how long will it
be before he has $150? If it is compounded continuously, how long will it be?
𝟏𝟓𝟎 = 𝟏𝟎𝟎 (𝟏 +
𝟎.𝟎𝟖 (𝟐𝒕)
𝟐
)
𝟏. 𝟓 = (𝟏. 𝟎𝟒)(𝟐𝒕)  𝐥𝐧𝟏. 𝟓 = (𝟐𝐭)𝐥𝐧𝟏. 𝟎𝟒
*divide by 100
2ln1.04
𝟏𝟓𝟎 = 𝟏𝟎𝟎𝐞(.𝟎𝟖𝒕) *divide by 100 𝟏. 𝟓 = 𝐞(.𝟎𝟖𝒕)  𝐥𝐧𝟏. 𝟓 = 𝟎. 𝟎𝟖𝐭
0.08
0.08
2ln1.04
t  5.17
t  5.07
5) Given 2 functions f and g, the composite function, denoted fog is defined by
fog(x) = f(g(x))
6) Change the exponential statement to an equivalent statement involving a logarithm:
7.4 = a6
loga7.4 = 6
7) Decide whether the following statement is true or false.
The graph of the exponential function f(x) = ax, a>0, a≠1, has no x-intercept.
8) Suppose that f(x) = 7x
a) What is f(5)? When x = 5, what is the point on the graph? 16807 (5, 16807)
1
1
b) If f(x) =
, what is x? When f(x) = , what is the point on the graph of f?
7x =
343
1
343
x = -3
(-3,
1
343
)
343
9) Solve the equation. Write the answer in terms of the common logarithm.
2 ∙ 101−x = 7
7
log10 101−x = log10 ( )
7
2
1 − 𝑥 = log( )
2
7
−𝑥 = log10 ( ) − 1
2
𝟕
𝒙 = 𝟏 − 𝐥𝐨𝐠 ( )
𝟐
10) The price p, of a specific used car that is x years old is given by the following
formula p(x) = 15310(0.89)x
a) How much does a 3-year old car cost? 15310(0.89)3  $10793.08
b) How much does a 9-year old car cost? 15310(0.89)9  $5363.96
11) Solve the equation: log5125 = 3x + 3
3 = 3x + 3  3x = 0  x = 0
12) If f is one-to-one function and f(1) = 5 then f-1(5) = 1
13) If f denotes the inverse of a function f-1, then the graphs of f and f-1 are symmetric
with respect to the line y = x
14) Determine the exponential function whose graph
f(x) = 2x
15) Solve the following logarithmic equation:
2log2(x-2) + log28 = 5
8(x-2)2 = 32
8x2 – 32x + 32 = 32
8(x-4) = 0
x = 4
3
16) Solve the equation: 3𝑥 = 243𝑥
3
3𝑥 = 35𝑥  x3 = 5x  x(x2 – 5)  x = −√5, 0, √5
17) Complete the sentence: loga1 = 0
18) If 5x = 7, what does 5-2x equal? 7−2 =
1
49
19) f(x) = 3-x+4
GRAPH reflects across y and shifts up 4 units
Domain (-,)
Range (4,)
Horizontal asymptote y = 4
20) logaMr = rlogaM
21) Solve the equation logx144 = 2
x2 = 144  x = 12
22) Solve the exponential equation: 36x - 46x = -4
u = 6x
U2 -4u + 4 (u-2)(u-2)
u = 2 then 6x = 2  log62  0.387
23) The population of a southern city follows the exponential law. If the population
doubled in size over 16 months and the current population is 30,000, what will the
population be 3 years from now?
𝑙𝑛2
Find rate (k) first: 60000 = 30000ek(16) 3 years  30000𝑒 36 16
(36 months)
2 = e16k
 142705 people
𝑙𝑛2
16
=k
7
24) Write the expression as a single logarithm: 4𝑙𝑜𝑔2 √6𝑥 − 7 − 𝑙𝑜𝑔2 ( ) + 𝑙𝑜𝑔2 7
𝑥
𝑙𝑜𝑔2 [𝑥 (6𝑥 − 7)2 ]
25) In working problems involving interest, if the payment period of the interest is
quarterly, then interest is paid 4 times a year.
26) Write the expression as a sum and differences of logarithms: 𝑙𝑜𝑔7 (
𝑥 15
𝑥−3
)
15𝑙𝑜𝑔7 𝑥 − 𝑙𝑜𝑔7 (𝑥 − 3)
27) Solve the following exponential equation. Express irrational solutions in exact form and
4 𝑥
as a decimal rounded to three decimal places ( ) = 91−𝑥
7
4
4
𝑙𝑛9
7
7
𝑙𝑛 7
𝑥𝑙𝑛 ( ) = (1 − 𝑥 )𝑙𝑛9  𝑥𝑙𝑛 ( ) + 𝑥𝑙𝑛9 = 𝑙𝑛9 𝑥 =
4
36
7
7
36
≈ 1.342
Condense 𝑙𝑛 ( ) + 𝑙𝑛9 = 𝑙𝑛 ( )
2
28) Solve the equation: 𝑒 𝑥 = 𝑒 19𝑥 ∙
1
𝑒 90
x2 = 19x-90  x2-19x+90=0
(x-9)(x-10)
29) Give the domain f(x) = ln(
1
𝑥−3
)
(3,)
x = 9,10
30)
f(x) = log(x-1) + 1
then GRAPH 
Domain is (1,)
Range is (-,)
Vertical asymptote x = 1
Find the inverse f -1(x) = 10x-1 + 1
then GRAPH 
Domain is (-,)
Range is (1,)
Vertical asymptote y = 1
31) Graph f(x) = 7x and f -1(x) = log7x

32) Write as a single logarithmic term:
𝑥
𝑥+5
𝑙𝑛
+ 𝑙𝑛
− ln(𝑥 2 − 25)
𝑥−5
𝑥
𝑥
𝑥+5
(
)(
)
𝑙𝑛 𝑥−52 𝑥
(𝑥 −25)
 𝑙𝑛
(
𝑥+5
)
𝑥−5
(𝑥−5)(𝑥+5)
𝑥+5
 ln(
𝑥−5
1
)((𝑥−5)(𝑥+5))  𝑙𝑛
1
(𝑥−5)2
-2ln(x-5)
33) Solve the following – give exact solution then round to 3 decimal places”
2(95x)=7
7
log 9 95x = log 9 ( )
2
7
5𝑥 = 𝑙𝑜𝑔9 ( )
7
𝑥=
log9 (2)
5
2
7
put in calculator as
𝑙𝑜𝑔(2)
5𝑙𝑜𝑔9
 0.114
34) Solve the following – give exact solution then round to 3 decimal places”
5x - 635-x =2
*multiply all terms by 5x to eliminate the 5 -x term
5x (5x - 635-x =2)
52x - 63 =25x u = 5x  u2 – 2u – 63 = 0
(u-9)(x+7)=0
u=9
5𝑥 = 9 then x = 𝑙𝑜𝑔5 9 put in calculator as
𝑙𝑜𝑔9
𝑙𝑜𝑔5
 1.365
35) Write as a single logarithmic term:
𝑙𝑜𝑔1 (𝑥 2 + 𝑥 ) − 𝑙𝑜𝑔1 (𝑥 2 + 𝑥 ) = −2
6
6
𝑙𝑜𝑔1/6
𝑥(𝑥+1)
𝑥(𝑥−1)
= −2 
(1⁄6)−2 = 36
𝑥+1
𝑥−1
=
36
1
 36x − 36 = x + 1 35𝑥 = 37
X =
37
35