Logarithmic Equations

In this lesson, we will look into how to solve logarithmic equations.

An equation that contains a logarithm of a variable quantity is called a logarithmic equation. Logarithmic equations can generally be solved using the properties of logarithm and the following property.

For two logarithms of the same base,

Log _a M = log _a N ↔ M = N

Example:

Solve the logarithmic equation
log ₂ (x – 1) + log ₂ (x – 4) = log ₂ (2x – 6)

Solution:

log ₂ (x – 1) + log ₂ (x – 4) = log ₂ (2x – 6)
log ₂ (x – 1)(x – 4) = log ₂ (2x – 6)
(x – 1)(x – 4) = (2x – 6)
x² – 5x + 4 = 2x – 6
x² – 7 x + 10 = 0
(x – 2)(x – 5) = 0
x = 2 or 5

We need to check whether each logarithm is defined for these values of x.

When x = 2,
log ₂ (x – 1) = log ₂ 1
log ₂ (x – 4) = log ₂ (– 2) , whuch is undefined
log ₂ (2x – 6) = log ₂ (– 2) , whuch is undefined
So, x = 2 is rejected

When x = 5,
log ₂ (x – 1) = log ₂ 4
log ₂ (x – 4) = log ₂ 1
log ₂ (2x – 6) = log ₂ 4
So, x = 5 is the answer

Example:

Solve the logarithmic equation
log ₃ 2 + log ₃ (x + 4) = 2log ₃ x

Solution:

log ₃ 2 + log ₃ (x + 4) = 2log ₃ x
log ₃ 2(x + 4) = log ₃ x²
2(x + 4) = x²
x² – 2x – 8 = 0
(x – 4)(x + 2) =0
x = 4 or –2

x = –2 is rejected because log ₃ (–2) is undefined
So the answer is x = 4

Videos

Properties of Logarithms - Part 1

Properties of Logarithms - Part 2 - Solving Logarithmic Equations

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