Log is an intermediate, not a delivery format
A log curve exists so a camera can hand a colourist as much of its sensor range as possible inside an ordinary file. It is deliberately low in contrast and desaturated, because contrast and saturation are decisions to be made later rather than baked in at capture.
That makes it the opposite of a finished picture. HDR10 is a delivery format describing how a graded image should appear on a display. Log describes what the sensor saw, awaiting interpretation.
Why converting it directly goes wrong
A conversion works on the image it is given. Given ungraded log, it sees a flat, low-contrast, desaturated picture and treats that as the intended look. What comes back is the same flat image with HDR signalling attached: technically an HDR file, visually still ungraded, and now harder to grade because it has been remapped.
The flatness is the point of log and it is also why log should never be the input to an automated conversion.
The lighter profiles are a different case
Manufacturers ship two tiers of log on many cameras: a full log curve and a lighter one, D-Log M against D-Log being the common example. The lighter curve keeps midtones nearer a normal contrast range while still protecting the highlights, which makes it far more survivable through a single automated conversion pass than full log is.
That is a real distinction rather than a hedge. If you are shooting specifically so that a clip can be converted without a grading stage in between, the lighter profile is the one to pick. Converting drone footage to HDR covers the profile choice per drone class, where this comes up most often.
What to do instead
Grade the log footage first, or at minimum apply the manufacturer's conversion LUT to bring it into a normal contrast range, then export. Convert the graded export.
If you are grading anyway and your tools support it, grading straight to an HDR timeline is better than grading to SDR and converting afterward: you keep the full sensor range through the whole chain instead of squeezing it into BT.709 and asking for some of it back.
Telling log from HDR in a file
Colour tags will not tell you, because log footage is usually tagged BT.709 or left untagged, exactly like ordinary SDR. The reliable signals are the camera and the picture profile recorded at capture, and the appearance itself: a flat, milky, desaturated image whose blacks sit well above zero is log.
Questions people ask
Which LUT should I apply first?
The manufacturer's own conversion LUT for your profile, which every major camera maker publishes: it maps their log curve into a normal contrast range without imposing a look. Apply it, export, then convert the export. A creative LUT is a separate decision made after that.
What if I have no grading tools at all?
Then shoot the lighter profile rather than full log. D-Log M and its equivalents sit close enough to a normal contrast curve to survive a single automated pass, which full log does not. If the footage already exists in full log, a free grading application is a smaller obstacle than the result of skipping the step.
Related: what conversion can and cannot do.