Properties of Logarithms

Logarithmic are the inverse of exponential functions. They are commonly used in many branches in math. Logarithms are written as f(x) = log _b (a) such that ‘b’ > 0, b = 1 and a > 0. This is read as “log base b of a”. Logarithmic functions have many properties and rule which are used to solve many questions:

General properties (where x> 0, y>0)

log_b(xy) = log_bx + log_by

log_b(x/y) = log_bx - log_by

log_b(x^m) = m log_b x

log_bb = 1

Example 1: Solve log_x 27 = 3, find the value of the base ‘x’.

Solution: The given equation is log_x 27 = 3

Convert this Logarithmic equation to Exponential equation by using the formula,

log_b(a) = N; a = b^N

Hence log_x 16 = 4 can be written as 27 = x³

Now we prime factorization of 27 = 3 * 3 * 3.

Therefore, 27 = 3³. This gives 27 = x³; 3³ = x³.

Hence x = 3 is the solution.

Example 2: Solve log_x 225 = 2, find the value of the base ‘x’.

Solution: The given equation is log_x 225 = 2.

Convert this Logarithmic equation to Exponential equation by using the formula,

log_b(a) = N; a = b^N

Hence log_x 225 = 2 can be written as 225 = x².

Now we prime factorization of 225 = 3 * 3 * 5 * 5.

Therefore, 225 = 15². This gives 225 = x²; 15² = x².

Hence x = 15 is the solution.


Solving Calculus Problems	Solve Calculus Problems	Range and Domain	Properties of Logarithmic Functions
Probability Formulas	Probability Questions	Probability Equations	Probabilities
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Properties of Logarithms

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