Solve 14/15*(7/17*5/28)^x=17

$$\frac { 14 } { 15 } \times ( \frac { 7 } { 17 } \times \frac { 5 } { 28 } ) ^ { x } = 17$$

Solve for x

$$x=\log_{\frac{5}{68}}\left(\frac{255}{14}\right)\approx -1.111926671$$

Steps Using Definition of Logarithm

Multiply both sides by \frac{15}{14}, the reciprocal of \frac{14}{15}.

$$\left(\frac{7}{17}\times \frac{5}{28}\right)^{x}=17\times \frac{15}{14}$$

Multiply 17 and \frac{15}{14} to get \frac{255}{14}.

$$\left(\frac{7}{17}\times \frac{5}{28}\right)^{x}=\frac{255}{14}$$

Multiply \frac{7}{17} and \frac{5}{28} to get \frac{5}{68}.

$$\left(\frac{5}{68}\right)^{x}=\frac{255}{14}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{5}{68}\right)^{x})=\log(\frac{255}{14})$$

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

$$x\log(\frac{5}{68})=\log(\frac{255}{14})$$

Divide both sides by \log(\frac{5}{68}).

$$x=\frac{\log(\frac{255}{14})}{\log(\frac{5}{68})}$$

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

$$x=\log_{\frac{5}{68}}\left(\frac{255}{14}\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(\frac{5}{68})}+\log_{\frac{5}{68}}\left(\frac{255}{14}\right)$$

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