Writing expressions as a single logarithm

Statement

• (a) Write as a single logarithm: $2\log x+3\log y-\log z$.
• (b) Write as a single logarithm, then compute its numerical value (base-$10$ logarithms): $\tfrac{1}{2}\log 49+\log 4-\log 14$.

Solution

(a) $2\log x+3\log y-\log z=\log\dfrac{x^2 y^3}{z}$.

(b) $\tfrac12\log 49=\log 7$, so $\log 7+\log 4-\log 14=\log\dfrac{7\cdot 4}{14}=\log 2$. In base $10$: $\log_{10}2\approx 0{.}3010$.

$\boxed{(a)\ \log\dfrac{x^2 y^3}{z};\qquad (b)\ \log 2\approx 0{.}3010}$