Nginx ignores config file option

I'm trying to start nginx with a specific config file as a user process on Ubuntu 14.04. However I'm getting stopped right at the start:

$ nginx -c nginx.conf 
nginx: [alert] could not open error log file: open() "/var/log/nginx/error.log" failed (13: Permission denied)
2017/03/30 01:03:46 [emerg] 21805#21805: open() "/usr/share/nginx/nginx.conf" failed (2: No such file or directory)

It seems to be continuing to try to open the default config and other files. I see nginx was build with these options:

$ nginx -V
nginx version: nginx/1.10.1
built with OpenSSL 1.0.1f 6 Jan 2014
TLS SNI support enabled
configure arguments: --with-cc-opt='-g -O2 -fPIE -fstack-protector --param=ssp-buffer-size=4 -Wformat -Werror=format-security -D_FORTIFY_SOURCE=2' --with-ld-opt='-Wl,-Bsymbolic-functions -fPIE -pie -Wl,-z,relro -Wl,-z,now' --prefix=/usr/share/nginx --conf-path=/etc/nginx/nginx.conf --http-log-path=/var/log/nginx/access.log --error-log-path=/var/log/nginx/error.log --lock-path=/var/lock/nginx.lock --pid-path=/run/nginx.pid --http-client-body-temp-path=/var/lib/nginx/body --http-fastcgi-temp-path=/var/lib/nginx/fastcgi --http-proxy-temp-path=/var/lib/nginx/proxy --http-scgi-temp-path=/var/lib/nginx/scgi --http-uwsgi-temp-path=/var/lib/nginx/uwsgi --with-debug --with-pcre-jit --with-ipv6 --with-http_ssl_module --with-http_stub_status_module --with-http_realip_module --with-http_auth_request_module --with-http_addition_module --with-http_dav_module --with-http_geoip_module --with-http_gunzip_module --with-http_gzip_static_module --with-http_image_filter_module --with-http_v2_module --with-http_sub_module --with-http_xslt_module --with-stream --with-stream_ssl_module --with-mail --with-mail_ssl_module --with-threads --add-module=/build/nginx-abUnII/nginx-1.10.1/debian/modules/nginx-auth-pam --add-module=/build/nginx-abUnII/nginx-1.10.1/debian/modules/nginx-dav-ext-module --add-module=/build/nginx-abUnII/nginx-1.10.1/debian/modules/nginx-echo --add-module=/build/nginx-abUnII/nginx-1.10.1/debian/modules/nginx-upstream-fair --add-module=/build/nginx-abUnII/nginx-1.10.1/debian/modules/ngx_http_substitutions_filter_module

Any idea how I can get nginx to start as a user process with the specified config file?

Obviously I can work around this problem by including the desired config at the system level and starting nginx normally as root and then switching the www-data or another user, but I'd like in this case to be able to run nginx as a user process no different than any other user application to fit my deployment architecture.