Laws of Logarithms (Edexcel International A Level (IAL) Maths: Pure 2): Exam Questions

By writing $1={\log }_{a}a$, show that

$1+2{\log }_{a}b+3{\log }_{a}c={\log }_{a}a{b}^2{c}^3.$

Solve the equation

$5^{2x}-25=0$

Solve the equation

$3^{2x-1}=4^3+4^2+1$

Solve the equation

${\log }_x(5x-6)=2.$

On the same axes, sketch the graphs of $y=2^x$ and $y=3^x$,
labelling any points where the graphs cross the coordinate axes and writing down the equation of any asymptotes.

Using a calculator, find to 3 significant figures:

(i) ${\log }_{2}5+{\log }_{5}2$

(ii) $\log 25-\ln 2$

(iii) $\log 200+{\log }_{5}50-\log 20+\ln 10$

Solve $3{\log }_{2}4+3x=5{\log }_{6}216$

Solve $2^{2x}-24\left(2^x\right)+128=0$.

Solve the equation

$4^{3x+2}={16}^{x+6}.$

Solve the equation

$4^{2x+3}-8=92$

giving your answer to 3 significant figures.

Solve the equation

$5^{2x}-8\times 5^x+12=0,$

giving your answers in the form ${\log }_{a}b$.

Solve the equation

${\log }_3(x+4)=4+2{\log }_{3}x$

giving your answers correct to 3 significant figures.

Solve the equation

$2{\log }_x(x+2)=3$

giving your answer correct to 3 significant figures.

On the same axes, sketch the graphs of $y=4^x$ and $y=5^x$.
Label any points of intersection with the coordinate axes.
Write down the equations of any asymptotes.

Write down an equation for the graph that is a reflection of $y=4^x$ in the $y$-axis.

Find the value of $\log 1000+\log 10 000$.

Write down the value of $a$ in the statement $6^{{\log }_{6}a}=36$.

Evaluate $\frac{2{\log }_{4}64+3^{{\log }_{2}8}-{\log }_{5}5}{\log 100}.$

Solve $2\log 1000=x{\log }_{16}4.$

Solve $3{\log }_{4}x={\log }_{4}x+3{\log }_{5}25.$

Solve $2\left(2^{2x}\right)+4=9\left(2^x\right).$

Solve the equation

$2\times 5^{2x+1}+21=41\times 5^x,$

giving your answers in the form ${\log }_{a}b$, where $a$ and $b$are rational numbers to be found.

Show that

$2{\log }_{3}x+{\log }_3(x^2-1)-2{\log }_3(x+1)\equiv {\log }_3\left(\frac{x^2(x-1)}{x+1}\right).$

Write the following as a single logarithm

$2{\log }_p(x+1)+3{\log }_p(x-1)-{\log }_p(x^2-1).$

Without using a calculator, show that

${\log }_{4}8={\log }_{9}27.$

On the same axes, sketch the graphs of $y=0.3^x$ and $y=0.5^x$.
Label any points of intersection with the coordinate axes.
Write down the equations of any asymptotes.

Write down an equation for the graph that is a reflection of $y=0.5^x$ in the $y$-axis.

Solve $3^{2(x+1)}+3=28\left(3^x\right).$

Find two values of $x$ for which $\log \left(x^2\right)=(\log x{)}^2$ is true.

Solve the equation ${\log }_4(2-x)={\log }_{16}(13-4x).$