Solve sqrt{2^99+2^99}=3^x

$$\sqrt{ { 2 }^{ 99 } + { 2 }^{ 99 } } = { 3 }^{ x }$$

Solve for x

$$x=50\log_{3}\left(2\right)\approx 31.546487679$$

Steps Using Definition of Logarithm

Calculate 2 to the power of 99 and get 633825300114114700748351602688.

$$\sqrt{633825300114114700748351602688+2^{99}}=3^{x}$$

Calculate 2 to the power of 99 and get 633825300114114700748351602688.

$$\sqrt{633825300114114700748351602688+633825300114114700748351602688}=3^{x}$$

Add 633825300114114700748351602688 and 633825300114114700748351602688 to get 1267650600228229401496703205376.

$$\sqrt{1267650600228229401496703205376}=3^{x}$$

Calculate the square root of 1267650600228229401496703205376 and get 1125899906842624.

$$1125899906842624=3^{x}$$

Swap sides so that all variable terms are on the left hand side.

$$3^{x}=1125899906842624$$

Take the logarithm of both sides of the equation.

$$\log(3^{x})=\log(1125899906842624)$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$x\log(3)=\log(1125899906842624)$$

Divide both sides by \log(3).

$$x=\frac{\log(1125899906842624)}{\log(3)}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$x=\log_{3}\left(1125899906842624\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(3)}+50\log_{3}\left(2\right)$$

$$n_{1}\in \mathrm{Z}$$