Solve 2^48=16^x

$${ 2 }^{ 48 } = { 16 }^{ x }$$

Solve for x

$$x=12$$

Steps Using Definition of Logarithm

Calculate 2 to the power of 48 and get 281474976710656.

$$281474976710656=16^{x}$$

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

$$16^{x}=281474976710656$$

Take the logarithm of both sides of the equation.

$$\log(16^{x})=\log(281474976710656)$$

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

$$x\log(16)=\log(281474976710656)$$

Divide both sides by \log(16).

$$x=\frac{\log(281474976710656)}{\log(16)}$$

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

$$x=\log_{16}\left(281474976710656\right)$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{2\ln(2)}+12$$

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