Solve {left(3/5right)}^15*3={left(5/3right)}^8x

$${ \left( \frac{ 3 }{ 5 } \right) }^{ 15 } \times 3= { \left( \frac{ 5 }{ 3 } \right) }^{ 8x }$$

Solve for x

$$x=\frac{\log_{\frac{5}{3}}\left(\frac{43046721}{30517578125}\right)}{8}\approx -1.606167487$$

Steps Using Definition of Logarithm

Calculate \frac{3}{5} to the power of 15 and get \frac{14348907}{30517578125}.

$$\frac{14348907}{30517578125}\times 3=\left(\frac{5}{3}\right)^{8x}$$

Multiply \frac{14348907}{30517578125} and 3 to get \frac{43046721}{30517578125}.

$$\frac{43046721}{30517578125}=\left(\frac{5}{3}\right)^{8x}$$

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

$$\left(\frac{5}{3}\right)^{8x}=\frac{43046721}{30517578125}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{5}{3}\right)^{8x})=\log(\frac{43046721}{30517578125})$$

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

$$8x\log(\frac{5}{3})=\log(\frac{43046721}{30517578125})$$

Divide both sides by \log(\frac{5}{3}).

$$8x=\frac{\log(\frac{43046721}{30517578125})}{\log(\frac{5}{3})}$$

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

$$8x=\log_{\frac{5}{3}}\left(\frac{43046721}{30517578125}\right)$$

Divide both sides by 8.

$$x=\frac{\ln(\frac{43046721}{30517578125})}{8\ln(\frac{5}{3})}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{4\ln(\frac{5}{3})}-\frac{\log_{\frac{5}{3}}\left(\frac{30517578125}{43046721}\right)}{8}$$

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