Look for the informational tag on your hairdryer, typically attached to the cord. This tag will tell you voltage range that is compatible with your hairdryer. Look for something similar to "110 - 240V" to show you that your hairdryer will work on the 240 volts in Australia. If your hairdryer will only work on North America voltage, your hairdryer will not operate in Australia even if you have a travel converter.
Purchase a travel converter for Australia. Australia uses the "Type 1" converter.
Plug your hairdryer into the travel converter, then plug your adapter into an Australian outlet. Your hairdryer should operate.
