Solve y=3cos(logx)+4sin(logx)

$$y = 3 \cos ( \log x ) + 4 \sin ( \log x )$$

Solve for x (complex solution)

$$x=10^{\left(-1\right)\left(i\ln(\left(\frac{3}{25}+\frac{4}{25}i\right)y+\left(-\frac{3}{25}-\frac{4}{25}i\right)\left(y^{2}-25\right)^{\frac{1}{2}})+\left(-2\right)n_{11}\pi \right)}\text{, }n_{11}\in \mathrm{Z}\text{, }Im(\ln(10^{\left(-1\right)\left(i\ln(\left(\frac{3}{25}+\frac{4}{25}i\right)y+\left(-\frac{3}{25}-\frac{4}{25}i\right)\left(y^{2}-25\right)^{\frac{1}{2}})+\left(-2\right)n_{11}\pi \right)}))+\ln(10)Re(\ln(\left(\frac{3}{25}+\frac{4}{25}i\right)y+\left(-\frac{3}{25}-\frac{4}{25}i\right)\left(y^{2}-25\right)^{\frac{1}{2}}))=0$$

$$x=10^{\left(-1\right)\left(i\ln(\left(\frac{3}{25}+\frac{4}{25}i\right)y+\left(\frac{3}{25}+\frac{4}{25}i\right)\left(y^{2}-25\right)^{\frac{1}{2}})+\left(-2\right)n_{12}\pi \right)}\text{, }n_{12}\in \mathrm{Z}\text{, }Im(\ln(10^{\left(-1\right)\left(i\ln(\left(\frac{3}{25}+\frac{4}{25}i\right)y+\left(\frac{3}{25}+\frac{4}{25}i\right)\left(y^{2}-25\right)^{\frac{1}{2}})+\left(-2\right)n_{12}\pi \right)}))+\ln(10)Re(\ln(\left(\frac{3}{25}+\frac{4}{25}i\right)y+\left(\frac{3}{25}+\frac{4}{25}i\right)\left(y^{2}-25\right)^{\frac{1}{2}}))=0$$

Solve for y (complex solution)

$$y=\left(\frac{3}{2}-2i\right)e^{i\log(x)}+\left(\frac{3}{2}+2i\right)e^{-i\log(x)}$$

$$x\neq 0$$

Solve for y

$$y=3\cos(\log(x))+4\sin(\log(x))$$

$$x>0$$