Logarithms — Practice
A logarithm is the exponent — nothing more, and nothing less.
What these questions cover
These questions use 6 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Integer — A whole number, positive, negative, or zero.
- Base — The number being multiplied. In 53 the base is 5.
- Exponent — How many times the base is used as a factor. In 53 the exponent is 3.
- Zeros — The x values where y = 0.
- Logarithm — logb y is the exponent that turns b into y.
- Quotient — The answer to a division.
Problem 1
Evaluate log2 8.
Worked answer
3
23=8
Problem 2
Evaluate log5 25.
Worked answer
2
52=25
Problem 3
Evaluate log10 1000.
Worked answer
3
103=1000
Problem 4
Evaluate log7 1.
Worked answer
0
70=1
Problem 5
Evaluate log4 4.
Worked answer
1
41=4
Problem 6
Rewrite 34=81 in logarithmic form. Give the value of log3 81.
Worked answer
4
log3 81 = 4
Problem 7
Evaluate log2 18.
Worked answer
-3
2-3=18
Problem 8
Write log(xy) as a sum. If log x=2 and log y=5, what is log(xy)?
Worked answer
7
log x+log y
2+5=7
Problem 9
If log x = 6 and log y = 2, find logxy.
Worked answer
4
6-2=4
Problem 10
If log x = 3, find log x4.
Worked answer
12
4log x = 4(3)=12
Problem 11
Write log 4 + log 25 as a single common logarithm and evaluate it.
Worked answer
2
log(4 × 25)=log 100
=2
Problem 12
Solve log3 x = 4.
Worked answer
81
x=34
=81
Problem 13
Solve 2x = 32.
Worked answer
5
32=25
x=5
Problem 14
Solve log2(x-1)=3.
Worked answer
9
x-1=23=8
x=9
Problem 15
Which statement is false?
- log(M+N)=log M+log N
- log(MN)=log M+log N
- log Mp = plog M
- logMN=log M-log N
Worked answer
log(M+N)=log M+log N
log2(4+4)=3 but log2 4+log2 4=4
There is no law for the log of a sum
Problem 16
For log(x-4) to be defined, what is the smallest integer x can be?
Worked answer
5
x-4>0
x>4, so the smallest integer is 5