Solve 100000000=67*1026^x

$$100000000 = 67 \times { 1026 }^{ x }$$

Solve for x

$$x=\log_{1026}\left(\frac{100000000}{67}\right)\approx 2.050356378$$

Steps Using Definition of Logarithm

Divide both sides by 67.

$$\frac{100000000}{67}=1026^{x}$$

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

$$1026^{x}=\frac{100000000}{67}$$

Take the logarithm of both sides of the equation.

$$\log(1026^{x})=\log(\frac{100000000}{67})$$

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

$$x\log(1026)=\log(\frac{100000000}{67})$$

Divide both sides by \log(1026).

$$x=\frac{\log(\frac{100000000}{67})}{\log(1026)}$$

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

$$x=\log_{1026}\left(\frac{100000000}{67}\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(1026)}+\log_{1026}\left(\frac{100000000}{67}\right)$$

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