Tutorial : Solving Equations Involving Logarithms Problem 1

Solve log₁₀5 – 1 = log₁₀(2x + 7) – log₁₀(x + 15).

Taken from the Integrated Curriculum for Secondary Schools Additional Mathematics Form 4 2005 ed. page 86 Practice 16 Question 1(d)

The laws of logarithms state that,
log_axy = log_ax + log_ay
log_ax/y = log_ax – log_ay
log_axⁿ = n log_ax

First, group all terms with log₁₀.
log₁₀5 – 1 = log₁₀(2x + 7) – log₁₀(x + 15)
log₁₀5 – log₁₀(2x + 7) + log₁₀(x + 15) = 1

We simplify log₁₀5 – log₁₀(2x + 7) + log₁₀(x + 15).
Use these laws of logarithms.
log_axy = log_ax + log_ay
log_ax/y = log_ax – log_ay
We get log₁₀5(x + 15) / (2x + 7) = 1.

Rearranging the equation, we get
5(x + 15) / (2x + 7) = 10¹
5x + 75 = 10 (2x + 7)
5x + 75 = 20x + 70
15x = 5.
x = 5/15 = 1/3

We can then conclude that x = 1/3.

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