Rearrange some expressions involving logarithms by applying the relation $\log_b(a) = c \iff a = b^c$.

Solve for $x$: $c(a^2)^x + d(a)^{x+1}=b$ (there is only one solution for this example).

Differentiate $\displaystyle \ln((ax+b)^{m})$

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Customised for the Numbas demo exam

Factorise $x^2+cx+d$ into 2 distinct linear factors and then find $\displaystyle \int \frac{ax+b}{x^2+cx+d}\;dx,\;a \neq 0$ using partial fractions or otherwise.

Video in Show steps.

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This quiz contains questions on algebraic fractions, logarithmic equations, exponential equations, quadratic equations and simultaneous equations.

Given a sum of logs, all numbers are integers,

$\log_b(a_1)+\alpha\log_b(a_2)+\beta\log_b(a_3)$ write as $\log_b(a)$ for some fraction $a$.

Also calculate to 3 decimal places $\log_b(a)$. 

Solve for $x$:  $\log_{a}(x+b)- \log_{a}(x+c)=d$

Solve for $x$: $\displaystyle 2\log_{a}(x+b)- \log_{a}(x+c)=d$. 

Make sure that your choice is a solution by substituting back into the equation.

Solve for $x$:  $\log_{a}(x+b)- \log_{a}(x+c)=d$

Solve for $x$: $\displaystyle 2\log_{a}(x+b)- \log_{a}(x+c)=d$. 

Make sure that your choice is a solution by substituting back into the equation.

Solve for $x$: $\log(ax+b)-\log(cx+d)=s$

Solve for $x$ each of the following equations: $n^{ax+b}=m^{cx}$ and $p^{rx^2}=q^{sx}$.

Solve for $x$: $c(a^2)^x + d(a)^{x+1}=b$ (there is only one solution for this example).

Express $\log_a(x^{c}y^{d})$ in terms of $\log_a(x)$ and $\log_a(y)$. Find $q(x)$ such that $\frac{f}{g}\log_a(x)+\log_a(rx+s)-\log_a(x^{1/t})=\log_a(q(x))$.

There is a video included explaining the rules of logarithms by going through simplification of logs of numbers rather than algebraic expressions.

Find $\displaystyle \int \frac{2ax + b}{ax ^ 2 + bx + c}\;dx$

Find $\displaystyle I=\int \frac{2 a x + b} {a x ^ 2 + b x + c}\;dx$ by substitution or otherwise.

Area and perimeter of parallelograms

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$f(x)= ae^{-bt}+c$ is given and plotted. A few points are plotted on the curve. $x$-coordinates are provided for two of them and $y$-coordinate provided for third. Student is required to determine other coordinates.

A question to test basic differentiation of functions, including powers of x, trig, log and exponential functions.