Solve x^sqrt(log x) = 10^125, where log is the base-10 logarithm.

Step-by-step solution

1. 1

   Substitute so that both sides become powers of 10. The obstacle is that $x$ appears in the base and inside the exponent. Setting

   $y=\log x\quad\Longleftrightarrow\quad x=10^{y}$

   replaces the base with a power of $10$ and the exponent with $\sqrt{y}$, matching the right-hand side's form.

2. 2

   Rewrite the equation with the substitution and use the power-of-a-power rule.

   $\left(10^{y}\right)^{\sqrt{y}}=10^{125}\quad\Longrightarrow\quad 10^{\,y\sqrt{y}}=10^{125}$

3. 3

   Equate the exponents. Since $t\mapsto 10^{t}$ is one-to-one, the exponents must be equal:

   $y\sqrt{y}=125\quad\Longrightarrow\quad y^{3/2}=125$

   Note $y\sqrt y=y^{1}\cdot y^{1/2}=y^{3/2}$, and $y\ge0$ is required for $\sqrt{\log x}$ to be real.

4. 4

   Solve the fractional-power equation. Raise both sides to the power $\tfrac23$, the reciprocal of $\tfrac32$:

   $y=125^{2/3}=\left(5^{3}\right)^{2/3}=5^{2}=25$

5. 5

   Undo the substitution. From $y=\log x=25$:

   $x=10^{25}$

6. 6

   Check the answer in the original equation. With $x=10^{25}$, $\log x=25$ and $\sqrt{\log x}=5$, so the left side is $\left(10^{25}\right)^{5}=10^{125}$ — exactly the right side. The exponent arithmetic $25\times5=125$ is the whole check.