Math · College algebra · Worked example
Solve a logarithmic equation and check it
Solve log₂(x + 1) + log₂(x − 1) = 3.
Combine the logarithms
The product rule turns the sum of logs into the log of a product: (x + 1)(x − 1) = x² − 1.
Rewrite in exponential form
log₂(x² − 1) = 3 means x² − 1 = 2³ = 8.
Solve the quadratic
x² = 9, so the candidates are x = 3 and x = −3.
Check each candidate
x = 3 gives log₂ 4 + log₂ 2 = 2 + 1 = 3, which works. x = −3 gives log₂(−2), which is undefined, so it is rejected. Combining the logs widened the domain, which is how the extra candidate crept in.
Result
x = 3. The candidate x = −3 is extraneous.
Your turn
Solve ln x + ln(x − 3) = ln 10.
Show the answer and explanation
x = 5.
Combine: ln(x² − 3x) = ln 10, so x² − 3x − 10 = 0 and (x − 5)(x + 2) = 0. x = −2 makes ln x undefined, so only x = 5 works: ln 5 + ln 2 = ln 10.
Keep exploring
Open the graph: the curve y = log₂(x + 1) + log₂(x − 1) exists only for x > 1, and it meets y = 3 once, at x = 3.
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