Logarithmic and exponential functions

Show that $y = k\mathrm{e}^{ax^2}$ where $k$ and $a$ are exact constants.

Find the value of $y$ when $x = 3$.

Find the values of $x$ when $y = 10$.

Given that $x > 1$, find the least number of terms for the sum of this progression to be greater than $43\ln(x^{24})$.

Given that the 25th term of this progression is equal to 408, find the exact value of $x$.

Find $y$ in terms of $x$.

Find the values of $x$ for which your equation is valid.

On the axes below, use these values to draw the straight-line graph of $\ln y$ against $\ln x$.

Use your graph to find the values of $A$ and $b$.

Draw a straight line graph for $\ln y$ against $x$.

Find the equation of the line in part (a) and hence find the values of $A$ and $k$. Give each value correct to 1 significant figure.

Find the value of $x$ for which $y = 17$.

$\log_2 x^2 + \log_{16} x = 18$

$\mathrm{e}^{2x+1} - 10\mathrm{e}^{-2x-1} = 3$

Solve the equation $5^{2y-1} = 6 \times 3^y$, giving your answer correct to 3 decimal places.

Solve the equation $\mathrm{e}^{2x} - 4 + 3\mathrm{e}^{-2x} = 0$, giving your answers in exact form.