Solve ln(1/2-sqrt{3})=ln(2+sqrt{3})

$$\ln ( \frac { 1 } { 2 - \sqrt { 3 } } ) = \ln ( 2 + \sqrt { 3 } )$$

Verify

Solution Steps

Rationalize the denominator of \frac{1}{2-\sqrt{3}} by multiplying numerator and denominator by 2+\sqrt{3}.

$$\log_{e}\left(\frac{2+\sqrt{3}}{\left(2-\sqrt{3}\right)\left(2+\sqrt{3}\right)}\right)=\log_{e}\left(2+\sqrt{3}\right)$$

Consider \left(2-\sqrt{3}\right)\left(2+\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\log_{e}\left(\frac{2+\sqrt{3}}{2^{2}-\left(\sqrt{3}\right)^{2}}\right)=\log_{e}\left(2+\sqrt{3}\right)$$

Square 2. Square \sqrt{3}.

$$\log_{e}\left(\frac{2+\sqrt{3}}{4-3}\right)=\log_{e}\left(2+\sqrt{3}\right)$$

Subtract 3 from 4 to get 1.

$$\log_{e}\left(\frac{2+\sqrt{3}}{1}\right)=\log_{e}\left(2+\sqrt{3}\right)$$

Anything divided by one gives itself.

$$\log_{e}\left(2+\sqrt{3}\right)=\log_{e}\left(2+\sqrt{3}\right)$$

Subtract \log_{e}\left(2+\sqrt{3}\right) from both sides.

$$\log_{e}\left(2+\sqrt{3}\right)-\log_{e}\left(2+\sqrt{3}\right)=0$$

Combine \log_{e}\left(2+\sqrt{3}\right) and -\log_{e}\left(2+\sqrt{3}\right) to get 0.

$$0=0$$

Compare 0 and 0.

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