Solve log_{2}1/64+log_{2}2sqrt{2}-log_{3}1/81

$$\log _ { 2 } \frac { 1 } { 64 } + \log _ { 2 } 2 \sqrt { 2 } - \log _ { 3 } \frac { 1 } { 81 }$$

Evaluate

$$\sqrt{2}-2\approx -0.585786438$$

Solution Steps

The base 2 logarithm of \frac{1}{64} is -6.

$$-6+\log_{2}\left(2\right)\sqrt{2}-\log_{3}\left(\frac{1}{81}\right)$$

The base 2 logarithm of 2 is 1.

$$-6+1\sqrt{2}-\log_{3}\left(\frac{1}{81}\right)$$

The base 3 logarithm of \frac{1}{81} is -4.

$$-6+1\sqrt{2}-\left(-4\right)$$

The opposite of -4 is 4.

$$-6+1\sqrt{2}+4$$

Add -6 and 4 to get -2.

$$-2+1\sqrt{2}$$

For any term t, t\times 1=t and 1t=t.

$$-2+\sqrt{2}$$