During installation you should change the permissions of the 'Logs' directory to allow your service account to write files. Use the account with the least privileges needed to perform your services function, generally the NETWORK SERVICE account.
You can do this from an install class on the service:
void Installer1_AfterInstall(object sender, InstallEventArgs e)
{
string myAssembly = Path.GetFullPath(this.Context.Parameters["assemblypath"]);
string logPath = Path.Combine(Path.GetDirectoryName(myAssembly), "Logs");
Directory.CreateDirectory(logPath);
ReplacePermissions(logPath, WellKnownSidType.NetworkServiceSid, FileSystemRights.FullControl);
}
static void ReplacePermissions(string filepath, WellKnownSidType sidType, FileSystemRights allow)
{
FileSecurity sec = File.GetAccessControl(filepath);
SecurityIdentifier sid = new SecurityIdentifier(sidType, null);
sec.PurgeAccessRules(sid); //remove existing
sec.AddAccessRule(new FileSystemAccessRule(sid, allow, AccessControlType.Allow));
File.SetAccessControl(filepath, sec);
}