Exponentials & Logs (DP IB Analysis & Approaches (AA): HL): Exam Questions

Let $x=\ln 15$ and $y=\ln 3.$ Write down the following expressions in terms of $x$ and $y$.

$\ln 5.$

$\ln 45$.

$\ln 135$.

Let $r=\log 2$ and $s=\log 12$. Write down the following expressions in terms of $r$ and $s$.

$\log 24.$

$\log 3$

$\log 72.$

It is given that ${\log }_{a}8=3$.

Find the value of ${\log }_{a}64$.

Find the value of $a.$

Find the value of ${\log }_{a^2}8.$

Let ${\log }_{b}3=x$ and ${\log }_{b}16=y$.

Find an expression for ${\log }_{b}9$ in terms of $x$.

Find an expression for ${\log }_{b}4$ in terms of $y.$

Find an expression for ${\log }_{b}48$ in terms of $x$ and $y.$

Show that $\frac{{\left(4-2\sqrt{x}\right)}^2}{8x}$ can be written as $2x^{-1}-2x^{-\frac{1}{2}}+\frac{1}{2}.$         

Given that $8\sqrt{2}=2^a,$ find the value of $a.$

The expression $\frac{x\left(2x^4-\sqrt{x}\right)}{x^2}$ can be written as $2x^a-x^b$, where $a,b\in \mathrm{\mathbb{Q}}$.

Find the value of $a$ and the value of $b$.

Solve the equation ${16}^x-3\left(4^{x+1}\right)=28$. Give your answer in the form $\frac{\ln a}{\ln b}$, where $a,b\in \mathrm{\mathbb{Z}}$.

Let $f\left(x\right)=5\ln \left(x-7\right)$.

Find the values of $x$ for which $f\left(x\right)$ is undefined.

The graph of $y=f\left(x\right)$ crosses the $x$-axis at the point $P$, with coordinates $\left(p,0\right)$.

Find the value of $p$.

Write the expression $3\ln 2-\ln 4$ in the form $\ln k$, where $k\in \mathrm{\mathbb{Z}}$.

Hence, or otherwise, solve $3\ln 2-\ln 4=-\ln x$.

Solve the equation $4^x-3\times 2^{x+2}=64$.

Given that $2^m=8$ and $2^n=16$, write down the value of $m$ and of $n$. 

Hence or otherwise, solve $8^{2x+1}={16}^{2x-3}$.

Solve $5\ln 2-\ln 8=-\ln x.$

Solve ${27}^{4x+2}={81}^{8x-3}.$

Let $f\left(x\right)=\ln \left(\frac{x}{3}-1\right)$.

Find the values of $x$ for which $f\left(x\right)$ is undefined.

The graph of $y=f\left(x\right)$ crosses the $x$-axis at the point A, with coordinates $\left(a,0\right)$.

Find the value of $a$.

Solve the equation ${49}^{p+4}={35}^{2p}$ for $p$. Express your answer in terms of $\ln 7$ and $\ln 5$.

It is given that $p={\log }_{124}125\times {\log }_{123}124\times {\log }_{122}123\times \dots \times {\log }_{5}6$.

Given that $p\in \mathrm{\mathbb{Z}}$, find the value of $p$.

Solve the equation  ${\log }_{6}3+{\log }_{6}2x=2-{\log }_{6}12$.

Solve the equation ${\log }_{9}x-{\log }_{9}2=2+{\log }_{9}5$.

Find the value of ${\log }_{4}32+{\log }_{4}8$.

Find the value of ${64}^{{\log }_{4}3}$.

Find the value of ${\log }_{8}2560-{\log }_{8}5.$

Find the value of ${64}^{{\log }_{8}5}.$

Solve the equation ${\log }_{5}x-{\log }_{5}4=4+{\log }_{5}2.$

Solve the equation ${216}^{k+2}={12}^{3k}$ for $k$. Express your answer in terms of $\ln 6$ and $\ln 2.$

Solve the equation $2\times {25}^x-30\times 5^{x-1}=1.$

It is given that $p={\log }_{242}243\times {\log }_{241}242\times {\log }_{240}241\times \dots \times {\log }_{3}4$.

Given that $p\in \mathrm{\mathbb{Z}}$, find the value of $p$.

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

Find the integer values of $x$ and $y$ for which $x+y{\log }_{343}6-10{\log }_{49}42=0$.

Find the integer values of $x$ and $y$ for which $x+y{\log }_{9}5+12{\log }_{27}15=0$.