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WAEC Mathematics 2023 Paper

Question : 40

Total: 50

Solve:

log_3 x + log_3 (x - 8) = 2

Solution:

Concept:

Use the product rule of logarithms, then convert the equation into a quadratic and check the domain of the logarithms.

Explanation:

Start with:

\log_3 x + \log_3 (x - 8) = 2

Apply the product rule:

\log_3 [x(x - 8)] = 2

Since

2 = \log_3 9

, write:

\log_3 [x(x - 8)] = \log_3 9

The bases are equal, so equate the arguments:

x(x - 8) = 9

Expand and rearrange:

x^2 - 8x - 9 = 0

Factorise:

(x - 9)(x + 1) = 0

So:

x = 9

x = -1

Check the domain: both

\log_3 x

and

\log_3 (x - 8)

require

x > 8

Therefore,

x = -1

is invalid.

Answer:

x = 9

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