Logarithm Formulas

A logarithm is a mathematical concept that provides an alternative way of expressing exponents.

• It represents the power to which a given base must be raised to obtain a specific number.
• It is especially useful when dealing with exponential relationships, as it simplifies complex calculations and makes them easier to handle.

The properties of logarithms are a set of rules that help simplify and solve logarithmic expressions by applying the laws of exponents and are given below.

Other Logarithm Formulas

Various other Logarithm Formulas are:

• \log_b\left(n^{\frac{1}{a}}\right) = \frac{1}{a}\,\log_b(n)
• log of 1 = log_a1 = 0
• log_aa = 1 (Identity rule)
• log_ba= log_bc =>  a= c (Equality rule)
• a^{log_ax} = x (Raised to log)

Also Check

• Natural Log
• Difference Between log and ln
• Log Table
• Laws of Logarithms

Solved Examples

Example 1: Solve log₂(x) = 4
Solution:

log₂(x) = 4
2⁴ = x
x = 16

Example 2: Solve log₂(8) = x
Solution:

log₂(8) = x

⇒ 2^x = 8
⇒ 2^x = 2³
⇒ x = 3

Example 3: Find the value of x if log₆(x - 3) = 1.
Solution:

log₆(x - 3) = 1

⇒ 6¹ = (x - 3)
⇒ x - 3 = 6
⇒ x = 9

Example 4: Find  x if log(x - 2) + log(x + 2) = log₂1
Solution: 

log(x - 2) + log(x + 2) = log₂1  

⇒ log(x - 2) + log(x + 2) = 0 [log(1) =0]
⇒ log[(x - 2)(x + 2)] = 0 [Product Rule]
⇒ (x - 2)(x + 2) = 1 [Antilog(0) = 1]
⇒ x² - 4 = 1
⇒ x² = 5
⇒ x = ±√5 [Log of Negative Number is Not Defined]
⇒ x = √5

Example 5: Find the value of log₉(59049).
Solution:

Given log₉ (59049) [9⁵= 59049]

= log₉(9)⁵
= 5.log₉(9) (identity rule i.e log_aa]
= 5

Example 6: Express log₁₀(5) + 1 in form of log₁₀x
Solution:

Given log₁₀(5) + 1

= log₁₀(5) + log₁₀10 [Identity Rule]
= log₁₀(5 × 10) [Product Rule]
= log₁₀50

Example 7: Find the value of x if log₁₀(x² - 15) = 1.
Solution:

log₁₀(x² - 15) = 1
log₁₀(x² - 15) = log₁₀10 [Identity Rule]

Applying Antilog,

⇒ (x² - 15) = 10
⇒ x² = 25
⇒ x = ±5