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2001 SMT/Algebra Problems/Problem 7

Revision as of 20:58, 19 March 2025 by Eevee9406 (talk | contribs) (Created page with "==Problem== If <math>\log_A B + \log_B A = 3</math> and <math>A<B</math>, find <math>\log_B A</math>. ==Solution== Let <math>x=\log_B A</math>; then <math>\frac{1}{x}+x=3</ma...")
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Problem

If $\log_A B + \log_B A = 3$ and $A<B$, find $\log_B A$.

Solution

Let $x=\log_B A$; then $\frac{1}{x}+x=3$. From here we find that $x^2-3x+1=0$, so $x=\frac{3\pm\sqrt{5}}{2}$. Technically both solutions are possible even if $A<B$ (because of values less than 1); however, we assume that we are considering $1<A<B$; then $x=\log_B A<1$, so $x=\boxed{\frac{3-\sqrt{5}}{2}}$.

~ eevee9406

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