Laws of Logarithms (Edexcel IAL Maths): Exam Questions

Evaluate

(i) $\mathrm{log}_{3}27$

(ii) $\mathrm{log}_{5}625$

(ii) $\mathrm{log}_{2}\frac{1}{4}$

(iv) $\mathrm{log}_{a}a$

Show that

$3\mathrm{log}_{a}4+2\mathrm{log}_{a}256=22\mathrm{log}_{a}2.$

Solve the equation

$\mathrm{log}_{x}16=2$

Solve the equation

$7^{2x-1}=343.$

Show that

$4 \log ( \frac{27}{16} ) = 12 \log 3 - 16 \log 2.$

Find the value of

(i) $3^{3}$

(ii) $4^{-2}$

(iii) $9^{0.5}$

Write down the value of $a$ in the following statements:

(i) $3^{a}=27$

(ii) $a^{\frac{1}{3}}=5$

(iii) $4a^{2}=64$

Write down the value of $a$ in the following statements:

(i) $\mathrm{log}_{3}a=4$

(ii) $\mathrm{log}_{a}216=3$

(iii) $\mathrm{log}_{2}128=a$

Solve the equation

$2^{x}=16$

Solve the equation equation$x^{2}-12x+27=0$

Hence, or otherwise, solve the equation $(3^{x})^{2}-12(3^{x})+27=0.$

Solve the equation

$2\mathrm{log}_{3}9=5x-6$

Use a calculator to find the value of (i) $5\mathrm{log}_{3}7$ (ii) $2\mathrm{log}_{2}3+3\mathrm{log}_{3}2$ giving your answers to four significant figures.

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

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

Solve the equation

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

Solve the equation

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

Solve the equation

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

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) $\mathrm{log}_{2}5+\mathrm{log}_{5}2$

(ii) $\mathrm{log}25-\mathrm{ln}2$

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

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

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

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

$5^{2x}-8\times5^{x}+12=0,$

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

Solve the equation

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

giving your answers correct to 3 significant figures.

Solve the equation

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

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 $\mathrm{log}1000+\mathrm{log}10000$.

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

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

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

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

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

Solve the equation

$2\times5^{2x+1}+21=41\times5^{x},$

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

Show that

$2 log_{3} x + log_{3} ( x^{2} - 1 ) - 2 log_{3} ( x + 1 ) \equiv log_{3} ( \frac{x^{2} ( x - 1 )}{x + 1} ).$

Write the following as a single logarithm

$2\mathrm{log}_{p}(x+1)+3\mathrm{log}_{p}(x-1)-\mathrm{log}_{p}(x^{2}-1).$

Without using a calculator, show that

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

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 $3^{2(x+1)}+3=28(3^{x}).$

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

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