Solve 7(4log(2)-log(3)-log(5))+5(2log(5)-3log(2)-log(3))+3(4log(3)-4log(2)-log(5))=log(2)

$$7(4 \log ( 2 ) - \log ( 3 ) - \log ( 5 ) )+5(2 \log ( 5 ) -3 \log ( 2 ) - \log ( 3) ) +3(4 \log ( 3 ) -4 \log ( 2 ) - \log ( 5 ) )= \log ( 2 )$$

Verify

Solution Steps

Subtract \log_{10}\left(2\right) from both sides.

$$7\left(4\log_{10}\left(2\right)-\log_{10}\left(3\right)-\log_{10}\left(5\right)\right)+5\left(2\log_{10}\left(5\right)-3\log_{10}\left(2\right)-\log_{10}\left(3\right)\right)+3\left(4\log_{10}\left(3\right)-4\log_{10}\left(2\right)-\log_{10}\left(5\right)\right)-\log_{10}\left(2\right)=0$$

Use the distributive property to multiply 7 by 4\log_{10}\left(2\right)-\log_{10}\left(3\right)-\log_{10}\left(5\right).

$$28\log_{10}\left(2\right)-7\log_{10}\left(3\right)-7\log_{10}\left(5\right)+5\left(2\log_{10}\left(5\right)-3\log_{10}\left(2\right)-\log_{10}\left(3\right)\right)+3\left(4\log_{10}\left(3\right)-4\log_{10}\left(2\right)-\log_{10}\left(5\right)\right)-\log_{10}\left(2\right)=0$$

Use the distributive property to multiply 5 by 2\log_{10}\left(5\right)-3\log_{10}\left(2\right)-\log_{10}\left(3\right).

$$28\log_{10}\left(2\right)-7\log_{10}\left(3\right)-7\log_{10}\left(5\right)+10\log_{10}\left(5\right)-15\log_{10}\left(2\right)-5\log_{10}\left(3\right)+3\left(4\log_{10}\left(3\right)-4\log_{10}\left(2\right)-\log_{10}\left(5\right)\right)-\log_{10}\left(2\right)=0$$

Combine -7\log_{10}\left(5\right) and 10\log_{10}\left(5\right) to get 3\log_{10}\left(5\right).

$$28\log_{10}\left(2\right)-7\log_{10}\left(3\right)+3\log_{10}\left(5\right)-15\log_{10}\left(2\right)-5\log_{10}\left(3\right)+3\left(4\log_{10}\left(3\right)-4\log_{10}\left(2\right)-\log_{10}\left(5\right)\right)-\log_{10}\left(2\right)=0$$

Combine 28\log_{10}\left(2\right) and -15\log_{10}\left(2\right) to get 13\log_{10}\left(2\right).

$$13\log_{10}\left(2\right)-7\log_{10}\left(3\right)+3\log_{10}\left(5\right)-5\log_{10}\left(3\right)+3\left(4\log_{10}\left(3\right)-4\log_{10}\left(2\right)-\log_{10}\left(5\right)\right)-\log_{10}\left(2\right)=0$$

Combine -7\log_{10}\left(3\right) and -5\log_{10}\left(3\right) to get -12\log_{10}\left(3\right).

$$13\log_{10}\left(2\right)-12\log_{10}\left(3\right)+3\log_{10}\left(5\right)+3\left(4\log_{10}\left(3\right)-4\log_{10}\left(2\right)-\log_{10}\left(5\right)\right)-\log_{10}\left(2\right)=0$$

Use the distributive property to multiply 3 by 4\log_{10}\left(3\right)-4\log_{10}\left(2\right)-\log_{10}\left(5\right).

$$13\log_{10}\left(2\right)-12\log_{10}\left(3\right)+3\log_{10}\left(5\right)+12\log_{10}\left(3\right)-12\log_{10}\left(2\right)-3\log_{10}\left(5\right)-\log_{10}\left(2\right)=0$$

Combine -12\log_{10}\left(3\right) and 12\log_{10}\left(3\right) to get 0.

$$13\log_{10}\left(2\right)+3\log_{10}\left(5\right)-12\log_{10}\left(2\right)-3\log_{10}\left(5\right)-\log_{10}\left(2\right)=0$$

Combine 13\log_{10}\left(2\right) and -12\log_{10}\left(2\right) to get \log_{10}\left(2\right).

$$\log_{10}\left(2\right)+3\log_{10}\left(5\right)-3\log_{10}\left(5\right)-\log_{10}\left(2\right)=0$$

Combine 3\log_{10}\left(5\right) and -3\log_{10}\left(5\right) to get 0.

$$\log_{10}\left(2\right)-\log_{10}\left(2\right)=0$$

Combine \log_{10}\left(2\right) and -\log_{10}\left(2\right) to get 0.

$$0=0$$

Compare 0 and 0.

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