Solve 1349=8700(0.945)^x-4-551

$$1349 = 8700 ( 0.945 ) ^ { x - 4 } - 551$$

Solve for x

$$x=-\log_{0.945}\left(\frac{46400000000}{8081268993}\right)\approx 30.895168573$$

Steps Using Definition of Logarithm

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

$$8700\times 0.945^{x-4}-551=1349$$

Add 551 to both sides of the equation.

$$8700\times 0.945^{x-4}=1900$$

Divide both sides by 8700.

$$0.945^{x-4}=\frac{19}{87}$$

Take the logarithm of both sides of the equation.

$$\log(0.945^{x-4})=\log(\frac{19}{87})$$

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

$$\left(x-4\right)\log(0.945)=\log(\frac{19}{87})$$

Divide both sides by \log(0.945).

$$x-4=\frac{\log(\frac{19}{87})}{\log(0.945)}$$

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

$$x-4=\log_{0.945}\left(\frac{19}{87}\right)$$

Add 4 to both sides of the equation.

$$x=\frac{\ln(\frac{19}{87})}{\ln(\frac{189}{200})}-\left(-4\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(0.945)}-\log_{0.945}\left(\frac{46400000000}{8081268993}\right)$$

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