Exercises - Exponentials and logarithms - c CNMiKnO PG

Exercises - Exponentials and logarithms - c CNMiKnO PG

c CNMiKnO PG -
Exercises - Exponentials and logarithms
1
Exercise 1. Solve the following equations and inequalities.
√
a) (0.25)3x−1 = ( 2)5−x ,
x
1
x
b) 16 x+3 = 4( 28 ) 2x+5 ,
e) 7−x − 3 · 7x+1 > 4,
d) (0.3)7−x ≤ (0.09)x+2 ,
c) 8x−2 < 24−2x ,
f) 22x+1 − 33 · 2x−1 + 4 = 0.
Exercise 2. Find intersection points of graphs of the following pairs of functions.
2
a) y = 8x
− 23
2
2
, y = 0.125x+ 3 ,
b) y = 2x
+5x+ 29
2
, y = 0.25−x
−4.5x−3.75
.
Exercise 3. Solve the following inequalities.
2
9 9x
a) ( 11
)
−11x
2
11x
≥ ( 11
9 )
c) 5x − 20 > 53−x ,
−9x
b) 5 · 4x + 2 · 25x ≥ 7 · 10x ,
,
d*) 32x + ( 21 )−x · 3x+1 − 22x+2 ≥ 0.
Exercise 4. Calculate exact values.
a) log2 4,
b) log2 2,
g) 2
log2 5
,
c) log3 3,
h) 3
log3 5
,
d) log 1,
e) log0.5 1,
5
5
i) log4 4 ,
f) log6 1,
j) log 10 .
Exercise 5. Solve the following equations.
a) log(3 − x)(x − 5) = log(x − 3) + log(5 − x),
b) log3 (4 · 3x−1 − 1) = 2x − 1,
c) log(5x2 + 2x − 1) − log(x + 2) = 1.
Exercise 6. Solve the following inequalities.
a) log(x − 4) + log x ≤ log 21,
2
b) 3log3 x + xlog3 x ≤ 162,
c) log(2x + x − 13) > x − x log 5,
d) log(x−2)
x−1
x−3
≥ 1.
Exercise 7. Which number is bigger?
a) log3 222 or log2 33,
b) π e or eπ .
Most exercises were taken from the script ”Matematyka - podstawy z elementami matematyki wyższej” issued by the
Gdańsk University of Technology publishing house.
Vocabulary - Exponentials and logarithms
• exponential function - funkcja wykladnicza
• exponent - wykladnik
• logarithmic function - funkcja logarytmiczna
• logarithm - logarytm
• loga x - the logarithm of x to the base a
• log3 x - the logarithm of x to third base
• log x - decimal logarithm of x
• ln x - natural logarithm of x