Basic Mathematics & Logarithm
Logarithmic equations and domains
Grade 11
Question:
<p>Number of real roots of equation <span class="math">\(\log_3 (x^2 + 4x + 3) = (x - 3)\)</span></p>
<p>(a) 0</p>
<p>(b) 1</p>
<p>(c) 2</p>
<p>(d) infinite</p>
Step-by-Step Solution
Key Concept: For a logarithmic equation to have solutions, the argument must be positive. Check the domain constraints carefully.
<p><strong>Solution:</strong></p><p>The equation <span class="math">$3^{\log_3 (x^2 + 4x + 3)} = (x - 3)$</span> is defined if <span class="math">$x^2 + 4x + 3 > 0$</span></p><p>This gives <span class="math">$(x + 1)(x + 3) > 0$</span>, so <span class="math">$x < -3$</span> or <span class="math">$x > -1$</span></p><p>The equation has no real roots satisfying the given domain constraint.</p>
Correct Answer: A