Solve 734=11.2(1.56)^x

$$734 = 11.2 ( 1.56 ) ^ { x }$$

Solve for x

$$x=\log_{1.56}\left(\frac{1835}{28}\right)\approx 9.405731081$$

Steps Using Definition of Logarithm

Divide both sides by 11.2.

$$\frac{734}{11.2}=1.56^{x}$$

Expand \frac{734}{11.2} by multiplying both numerator and the denominator by 10.

$$\frac{7340}{112}=1.56^{x}$$

Reduce the fraction \frac{7340}{112} to lowest terms by extracting and canceling out 4.

$$\frac{1835}{28}=1.56^{x}$$

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

$$1.56^{x}=\frac{1835}{28}$$

Take the logarithm of both sides of the equation.

$$\log(1.56^{x})=\log(\frac{1835}{28})$$

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

$$x\log(1.56)=\log(\frac{1835}{28})$$

Divide both sides by \log(1.56).

$$x=\frac{\log(\frac{1835}{28})}{\log(1.56)}$$

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

$$x=\log_{1.56}\left(\frac{1835}{28}\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(1.56)}+\log_{1.56}\left(\frac{1835}{28}\right)$$

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