Logarithm Examples and Practice Problems

Take a real number x and b^x represents an unique real number. If we write a = b^x, then the exponent x is the logarithm of a with log base of b and we can write a = b^x as
log_ba = x
The notation x = log_ba is called Logarithm Notation.
Before goto the example look at this logarithm rules and logarithm calculator.

Example Logarithm Notations:
(i) 3 = log₄64 is equivalent to 4³ = 64
(ii) 1/2 = log₉3 is equivalent to √9 = 3

Logarithm Examples
1. Change the below lagarithm log₂₅5 = 1/2 to exponential form
log₂₅5 = 25^1/2 = 5

2. Change 1/8 = 8^-1 to logarithmic notation.
Solution:
1/8 = 8^-1 = log₈8

3. Evaluate the value for log₄8
Solution:
x = log₄8
Then we can say
4^x = 8 = 2³ ,taking square root both side
2^x = 2^3/2
x = 3/2

4. Find the value of x from log_x 100 = 2
solution:
log_b 1000 = 3
We can write it as,
b³ = 1000
b³ = 10³
So from the above equation
b = 10
5. Find the value for log₅8 + ₅(1/1000)
Solution:
log₅8 + ₅(1/1000) = loglog₅(8 x 1/1000)
= log₅(1/125)
= log₅(1/5)³
= log₅(5)^-3
= -3log₅(5)
= -3 x 1
= -3

6. Solve 2log₅3 X log₉x + 1 = log₅3
Solution:
We can rewrite the above equation in the below format
=> log₅3² X log₉x = log₅3 - 1
=> log₅9 X log₉x = log₅3 - log₅5
=> log₅9 X log₉x = log₅(3/5)
By Using the change of base rule in Left side we get
=> log₅x = log₅(3/5)
The value of X = 3/5

Logarithm Practice Problems:
1. Find Logarithm for following equations
(i) log₅125
(ii) log₂(2√2)
(iii) log_1/327

2. Find the value of x in following equations
(i) log_x 0.001 = -2
(ii) √log₂x = 3
(iii) 2 log₉ x = 1

3. Simplify the following expression into single term
(i) log₁₀2 + log₁₀10
(ii) 3log₃2 + 4 - 8 log₃3
(iii) log₁₀5 + log₁₀ 20 - log₁₀24 + log₁₀ 25 - 4

4. Solve the following logarithm equation
(i) 3log₅2 = 2log₅x
(ii) xlog₁₆8 = -1
(iii) log₅(10 + x) = log₅(3 + 4x)