Log time() WordPress user session started by wp_login(); and ended with wp_logout();

log_wp_login.php

// Log session login time when wp_login() is fired.
function user_session_start( $user_login, $user ) {
    update_user_meta( $user->ID, 'last_login', time() );
    return $user_login;
}
add_action( 'wp_login', 'user_session_start', 10, 2 );

log_wp_logout.php

// Log session logout time when wp_logout() is fired.
function user_session_logout($user_id){
  update_user_meta( $user_id, 'last_logout', time() );
} 
add_action( 'wp_logout', 'user_session_logout', 10, 1 ); 

// Note:  WordPress core does not have a way to log every possible type of session termination. 
// The user_session_logout hook will not let us log sessions ended by an auth cookie timeout or 
// other failed session checks.