# Find the equivalent Cartesian equation of the polar equations {eq}r\sin(\theta)=\ln(r)+\ln(\cos(\theta)) {/eq}.

## Question:

Find the equivalent Cartesian equation of the polar equations {eq}r\sin(\theta)=\ln(r)+\ln(\cos(\theta)) {/eq}.

## Polar Equation with Logarithmic Function:

If the polar equation is written with logarithmic functions and we need an equivalent cartesian equation, then we'll change the polar coordinates {eq}(r, \theta) {/eq} using the general formulas below. After that, apply the quotient property of logarithmic in the obtained expression.

{eq}\begin{align*} x&=r\cos\theta\\[2ex] y&=r\sin\theta\\[2ex] r&=\sqrt{x^2+y^2}\\[2ex] \ln\left ( \dfrac{p}{q} \right )&=\ln(p)-\ln(q)\\[2ex] \end{align*} {/eq}

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Given:

{eq}r\sin(\theta)=\ln(r)+\ln(\cos(\theta))\\[2ex] {/eq}

The formulas for the cartesian coordinates are:

{eq}x=r\cos\theta\\[2ex] y=r\sin\t...