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Q40E

Expert-verifiedFound in: Page 217

Book edition
7th

Author(s)
Kenneth H. Rosen

Pages
808 pages

ISBN
9780073383095

**Show that for all real numbers aand b with a>1 and b>1, if f(x) is O (log _{b }x), then f(x) is O (log _{a }x).**

Hence we conclude f(x) is O (log _{b} x), then f(x) is O (log _{a} x)

Given, f(x) is O (log _{b} x)

By the definition of Big-O notation, there exist a positive real number M ∋,

│f(x)│≤M│g(x)│, whenever x>k

Consider,

│f(x)│≤M │log _{b} x │

=M$\left|\frac{{\mathrm{log}}_{a}x}{{\mathrm{log}}_{b}x}\right|$

** **=** $=\frac{M}{{\mathrm{log}}_{b}x}\left|\mathrm{log}ax\right|$**

** **Let M_{1}=$\frac{M}{{\mathrm{log}}_{b}x}$** **, then

** **│f(x)│≤M_{1} │log _{a} x │ whenever x>k

Therefore by the definition of Big-O notation, f(x) is O (log _{a} x) with constant M and k.

**Final answer **

Hence we conclude f(x) is O (log _{b} x), then f(x) is O (log _{a} x)

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