Solve 75^107=2^x

$${ 75 }^{ 107 } = { 2 }^{ x }$$

Solve for x

$$x=107\log_{2}\left(75\right)\approx 666.483599883$$

Steps Using Definition of Logarithm

Calculate 75 to the power of 107 and get 428109813197592092360009829516938560023702667508500023874986287773100691815799870222979365956203869690474475197531367201813742129450940715669117140758306663014635951025610438591684214770793914794921875.

$$428109813197592092360009829516938560023702667508500023874986287773100691815799870222979365956203869690474475197531367201813742129450940715669117140758306663014635951025610438591684214770793914794921875=2^{x}$$

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

$$2^{x}=428109813197592092360009829516938560023702667508500023874986287773100691815799870222979365956203869690474475197531367201813742129450940715669117140758306663014635951025610438591684214770793914794921875$$

Take the logarithm of both sides of the equation.

$$\log(2^{x})=\log(428109813197592092360009829516938560023702667508500023874986287773100691815799870222979365956203869690474475197531367201813742129450940715669117140758306663014635951025610438591684214770793914794921875)$$

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

$$x\log(2)=\log(428109813197592092360009829516938560023702667508500023874986287773100691815799870222979365956203869690474475197531367201813742129450940715669117140758306663014635951025610438591684214770793914794921875)$$

Divide both sides by \log(2).

$$x=\frac{\log(428109813197592092360009829516938560023702667508500023874986287773100691815799870222979365956203869690474475197531367201813742129450940715669117140758306663014635951025610438591684214770793914794921875)}{\log(2)}$$

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

$$x=\log_{2}\left(428109813197592092360009829516938560023702667508500023874986287773100691815799870222979365956203869690474475197531367201813742129450940715669117140758306663014635951025610438591684214770793914794921875\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(2)}+107\log_{2}\left(75\right)$$

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