IB Mathematics Analysis and Approaches Question Type 4: Using Laws of Logarithms Exercises

Solve for $x$: ${\log }_2(x)+{\log }_2(x-3)=3$.

Given $a={\log }_{2}3$ and $b={\log }_{2}5$, express ${\log }_2(45)$ in terms of $a$ and $b$.

Given $a={\log }_{2}3$ and $b={\log }_{2}5$, find ${\log }_2\sqrt{15}$ in terms of $a$ and $b$.

Simplify the expression ${\log }_{10}\left(\frac{100x^2}{\sqrt{x}}\right)$. [4 marks]

Solve for $x$: ${\log }_3(2x+1)-{\log }_3(x-1)=2$.

Solve for $x$: $3{\log }_4(x)-{\log }_4(2)=2$.

Expand the logarithm $\ln \left(\frac{x^2\sqrt{y^3}}{z^4}\right)$ into a sum and difference of simpler ln terms. [3 marks]

Given ${\log }_3(b)=4$, find ${\log }_3\left(\frac{b}{9}\right)$.

Given $a={\log }_{5}2$ and $b={\log }_{5}3$, express ${\log }_{5}72$ in terms of $a$ and $b$. [3]

Solve for $x$: $\ln (x)+\ln (x-2)-\ln (3x)=0$.

Given ${\log }_{5}2=p$ and ${\log }_{5}7=q$, express ${\log }_5\frac{14}{25}$ in terms of $p$ and $q$.

If ${\log }_3(b)=4$, calculate ${\log }_3(1/b)$.

Given $a={\log }_{3}2$ and $b={\log }_{3}5$, express ${\log }_{3}40$ in terms of $a$ and $b$.

If ${\log }_3(b)=4$, determine ${\log }_3(27b)$.

Given $x,y,z>0$, write the expression $\log \left(\frac{\sqrt{x{y}^2}}{z^3}\right)$ in terms of $\log x$, $\log y$ and $\log z$.

Simplify the expression ${\log }_3(81)+2{\log }_3(3)-{\log }_3(9)$.

Given $a={\log }_{2}3$ and $b={\log }_{2}7$, express ${\log }_2\frac{21}{2}$ in terms of $a$ and $b$.

Solve for $x$: ${\log }_5(x^2-5x)={\log }_5(6)$.

Given ${\log }_3(b)=4$, find ${\log }_3\left(\sqrt{b}\right)$.

Given $a={\log }_{2}8$ and $b={\log }_{2}32$, express ${\log }_{2}16$ in terms of $a$ and $b$.

Simplify the expression ${\log }_5(125)+2{\log }_5\left(\frac{1}{5}\right)$.

Solve for $x$: ${\log }_3(x-1)+{\log }_3(x+2)=2$.

Write the expression $2\ln (a)+3\ln (b)-\ln (c)$ as a single logarithm.