Solve 1000000=784^x

$$1000000 = 78 { 4 }^{ x }$$

Solve for x

$$x = \frac{\log_{2} {(\frac{500000}{39})}}{2} \approx 6.823083175$$

Steps Using Definition of Logarithm

Divide both sides by 78.

$$\frac{1000000}{78}=4^{x}$$

Reduce the fraction \frac{1000000}{78} to lowest terms by extracting and canceling out 2.

$$\frac{500000}{39}=4^{x}$$

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

$$4^{x}=\frac{500000}{39}$$

Take the logarithm of both sides of the equation.

$$\log(4^{x})=\log(\frac{500000}{39})$$

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

$$x\log(4)=\log(\frac{500000}{39})$$

Divide both sides by \log(4).

$$x=\frac{\log(\frac{500000}{39})}{\log(4)}$$

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

$$x=\log_{4}\left(\frac{500000}{39}\right)$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(2)}+\frac{\log_{2}\left(\frac{500000}{39}\right)}{2}$$

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