# Some logarithmic functional equations

## Vichian Laohakosol, Watcharapon Pimsert, Charinthip Hengkrawit, and Bruce Ebanks

Address:

V. Laohakosol, W. Pimsert Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok 10900 and Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Ch. Hengkrawit Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathum Thani 12120, Thailand

B. Ebanks Department of Mathematics and Statistics, Mississippi State University, P.O. Drawer MA Mississippi State, MS 39762, U.S.A.

E-mail:

fscivil@ku.ac.th

fsciwcrp@ku.ac.th

charinthip@mathstat.sci.tu.ac.th

ebanks@math.msstate.edu

Abstract: The functional equation $f(y-x) - g(xy) = h\left(1/x-1/y\right)$ is solved for general solution. The result is then applied to show that the three functional equations $f(xy)=f(x)+f(y)$, $f(y-x)-f(xy)=f(1/x-1/y)$ and $f(y-x)-f(x)-f(y)=f(1/x-1/y)$ are equivalent. Finally, twice differentiable solution functions of the functional equation $f(y-x) - g_1(x)-g_2(y) = h\left(1/x-1/y\right)$ are determined.

AMSclassification: primary 39B20.

Keywords: logarithmic functional equation, Pexider equations.

DOI: 10.5817/AM2012-3-173