Logarithms Review and Practice Questions
- Posted by Brian Stocker MA
- Date June 20, 2014
- Comments 1 comment
A Logarithm is a quantity representing the power to which a fixed number (the base) must be raised to produce a given number.
Logarithms – a Quick Review
Logarithm is a function that has the form
logy x = a
It actually solves this equation: which number do we put as a degree on the variable y to get the variable x, that is:
y a = x
y is called the base and a is the exponent.
For example, let’s solve logarithm log525 = a.
Here, we represent 25 using 5 and the second degree. a and 2 are both on the number 5, so they must be the same.
5a = 25
5a = 52
Here, we represent 25 using 5 and the second degree. a and 2 are both on the number 5, so they must be the same.
a = 2
We can see from the way the logarithm works, that:
loga1 = 0 and logaa = 1
From loga1 = 0 we have that a0 = 1, which is true for any real number a.
From logaa = 1 we have that a1 = a, which is true for any real number a.
If in the logarithm the base is 10, then instead of log10 we write lg.
When we are solving some logarithm, any part can be unknown. In the first example, we had a case where the exponent was the unknown variable. Let’s see another example, where both exponent and base are known:
lg x = 2
102 = x
x = 100
Practice Questions
1. If log2x = 3, then x is:
A. 9
B. 8
C. 7
D. 6
2. Solve the equation log41/4 = x.
A. -1
B. 0
C. 1
D. 2
3. For what x is the following equation correct:
logx125 = 3
A. 1
B. 2
C. 3
D. 5
4. Find x if logx(9/25) = 2.
A. 3/5
B. 5/3
C. 6/5
D. 5/6
5. Solve log1010,000 = x.
A. 2
B. 4
C. 3
D. 6
6. Find x if log1/2 x = 4.
A.16
B. 8
C. 1/8
D. 1/16
Answer Key
1. B. 8
log2x = 3
23 = x
x = 8
2. A. -1
log41/4 = x.
4x = 1/4
4x = 4-1
x = -1
3. E. 5
logx125 = 3
x3 = 125
x3 = 53
x = 5
4. A. 3/5
logx(9/25) = 2
x2 = 9/25
x2 = (3/5)2
x = 3/5
5. B. 4
log1010,000 = x
10x = 10,000
10x = 104
x = 4
6. D. 1/16
log1/2 x = 4
(1/2)4 = x
x = 1/16
Date Published: Friday, June 20th, 2014
Date Modified: Friday, March 3rd, 2023
1 Comment
Thanks!