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Logarithms Equation Example

Question:

Given: lg(x + 3)³ - lg²(x + 3) - 2 = 0, What is the value of x?

Solution:

lg(x + 3)³ - lg²(x + 3) - 2 = 0

lg²(x + 3) - lg(x + 3)³ + 2 = 0

lg²(x + 3) - 3 lg(x + 3) + 2 = 0

let t = lg(x + 3), then lg²(x + 3) - 3 lg(x + 3) + 2 = 0 change to:

t² - 3t + 2 = 0

(t - 2) (t - 1) = 0

t₁ = 2 and t₂ = 1

substitute the value of t to the equation t = lg(x + 3)

equation1:

2 = lg(x + 3)

10² = x + 3

x = 10² - 3

x = 97

equation2:

1 = lg(x + 3)

10¹ = x + 3

x = 10 - 3

x = 7

The solution is: x₁ = 7 and x₂ = 97

Formula used in this example:

log_ab^x = x log_ab

If a^x = b (a > 0 and a is not equal to 1), then x = log_ab

when a = 10, log₁₀b = lg b