Find domain of 10^x+10^y =10.

10^x + 10^y = 10

⇒ 10^y = 10 – 10^x

⇒ 10^y = 10 (1 – 10^{x – 1})

⇒ log₁₀ 10^y = log₁₀ 10 + log₁₀ (1 – 10^{x – 1})

⇒ y = 1 + log₁₀ (1 – 10^{x – 1})

Now, for y to be defined log₁₀ (1 – 10^{x – 1}) must be defined.

i.e., 1 – 10^{x – 1} > 0

⇒ 1 > 10^{x – 1}

⇒ 10 > 10^x

i.e. 10^x < 10¹

⇒ x < 1

∴ Domain: (– ∞, 1)