Expanding a base 10 logarithm of a quotient

Quotient property: the log of a fraction is the log of the top minus the log of the bottom. $${\log }_{10}\left(\frac{kz}{w^2}\right)={\log }_{10}(kz)-{\log }_{10}(w^2)$$ Product property: the log of a product is the sum of the logs. $${\log }_{10}(kz)={\log }_{10}k+{\log }_{10}z$$ Power property: an exponent inside comes out front as a multiplier. $${\log }_{10}(w^2)=2{\log }_{10}w$$ Put together: $${\log }_{10}k+{\log }_{10}z-2{\log }_{10}w$$ That is choice B.

A numerical check with $k=10$, $z=100$, $w=10$: the left side is ${\log }_{10}\left(\frac{1000}{100}\right)={\log }_{10}10=1$, and choice B gives $1+2-2(1)=1$.

Each choice

• A. Two slips at once. The log of a product became the log of a sum, and $w^2$ was read as $2w$. Neither is an identity: with $k=z=w=10$ this choice gives ${\log }_{10}20-{\log }_{10}20=0$, not 1.
• B (correct). The full expansion: the quotient, product and power properties give ${\log }_{10}k+{\log }_{10}z-2{\log }_{10}w$.
• C. The exponent came down as $\frac{1}{2}$ instead of 2. That would be right for a square root, $\sqrt{w}=w^{1/2}$, not for $w^2$.
• D. The signs ended up on the wrong terms. $z$ is multiplied in the numerator, so its log is added, and $w^2$ is in the denominator, so its log is subtracted.