Solve 3.6/17.88={left(0.0384/0.0769right)}^x

$$\frac{ 3.6 }{ 17.88 } = { \left( \frac{ 0.0384 }{ 0.0769 } \right) }^{ x }$$

Solve for x

$$x=\log_{\frac{384}{769}}\left(\frac{30}{149}\right)\approx 2.307945249$$

Steps Using Definition of Logarithm

Expand \frac{3.6}{17.88} by multiplying both numerator and the denominator by 100.

$$\frac{360}{1788}=\left(\frac{0.0384}{0.0769}\right)^{x}$$

Reduce the fraction \frac{360}{1788} to lowest terms by extracting and canceling out 12.

$$\frac{30}{149}=\left(\frac{0.0384}{0.0769}\right)^{x}$$

Expand \frac{0.0384}{0.0769} by multiplying both numerator and the denominator by 10000.

$$\frac{30}{149}=\left(\frac{384}{769}\right)^{x}$$

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

$$\left(\frac{384}{769}\right)^{x}=\frac{30}{149}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{384}{769}\right)^{x})=\log(\frac{30}{149})$$

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

$$x\log(\frac{384}{769})=\log(\frac{30}{149})$$

Divide both sides by \log(\frac{384}{769}).

$$x=\frac{\log(\frac{30}{149})}{\log(\frac{384}{769})}$$

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

$$x=\log_{\frac{384}{769}}\left(\frac{30}{149}\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(\frac{384}{769})}+\log_{\frac{384}{769}}\left(\frac{30}{149}\right)$$

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