A-Level CAIE Mathematics A2 3.2 Logarithmic and exponential functions Question Bank

Use logarithms to solve the equation $4^{3 x-1}=3\left(5^{x}\right)$, giving your answer correct to 3 decimal places.

Given that $2^{y}=9^{3 x}$, use logarithms to show that y=k x and find the value of k correct to 3 significant figures.

Given that $\ln \left(1+\mathrm{e}^{2 y}\right)=x$, express y in terms of x.

Use logarithms to solve the equation $5^{2 x-1}=2\left(3^{x}\right)$, giving your answer correct to 3 significant figures.

Using the substitution $u=\mathrm{e}^{x}$, or otherwise, solve the equation

giving your answer correct to 3 significant figures.

The variables x and y satisfy the equation $y=4^{2 x-a}$, where a is an integer. As shown in the diagram, the graph of $\ln y$ against x is a straight line passing through the point (0,-20.8), where the second coordinate is given correct to 3 significant figures.

Solve the equation $\log _{10}(x+9)=2+\log _{10} x$.

Solve the equation $\ln \left(x^{3}-3\right)=3 \ln x-\ln 3$. Give your answer correct to 3 significant figures.

Solve the equation $\frac{3^{x}+2}{3^{x}-2}=8$, giving your answer correct to 3 decimal places.

Showing all necessary working, solve the equation $5^{2 x}=5^{x}+5$. Give your answer correct to 3 decimal places.