Logarithms

Part of the Oxford MAT Livestream.

Solutions are available here.

MAT syllabus

Laws of logarithms and exponentials. Solution of the equation $a^x = b$.

Revision

$a^ma^n=a^{m+n}$ for any positive real number $a$ and any real numbers $m$ and $n$.
$(a^m)^n=a^{mn}$ for any positive real number $a$ and any real numbers $m$ and $n$.
$\displaystyle a^{-n}=\frac{1}{a^n}$ for any positive real number $a$ and any real number $n$.
$a^0=1$ for any non-zero real number $a$.
The solution to $a^x=b$ where $a$ and $b$ are positive numbers (with $a\neq 1$) is $\log_a (b)$. In this expression, the number $a$ is called the base of the logarithm.
$\log_a (x)$ is a function of $x$ which is defined when $x>0$. Like with $\sin x$, sometimes the brackets are omitted if it's clear what the function is being applied to, so we might write $\log_a x$.
$\log_a x$ doesn't repeat any values; if $\log_a x=\log_a y$ then $x=y$.
Note the special case $\log_a a =1$ because $\log_a a$ is the solution $x$ to the equation $a^x=a$, and that solution is 1.
In fact, $\log_a (a^x)=x$.
In that sense, the logarithm function is the inverse function for $y=a^x$.
$a^{\log_a x}=x$.
$\log_a (xy)=\log_a(x)+\log_a(y)$.
$\log_a (x^k)=k\log_a x$ including $\displaystyle\log_a \frac{1}{x}=-\log_a x$.
There's a mathematical constant called $e$, which is just a number (it's about 2.7).
$e^x$ is called the exponential function.
The laws of indices and laws of logarithms above hold when the base $a$ is equal to $e$.
$\log_e x$ is sometimes written as $\ln x$ and the function is sometimes called the natural logarithm.

Revision Questions

Simplify $(2^3)^4$ and $(2^4)^3$ and $2^42^3$ and $2^32^4$.
Solve $x^{-2}+4x^{-1}+3=0$.
Simplify $\log_{10} 3+\log_{10} 4$ into a single term.
Write $\log_3(x^2+3x+2)$ as the sum of two terms, each involving a logarithm.
Solve $\log_x (x^2)=x^3$.
Solve $\log_x (2x)=3$ for $x>0$.
Solve $\log_{x+5}(6x+22)=2$.
Let $a=\ln 2$ and $b=\ln 5$, and write the following in terms of $a$ and $b$.
\begin{equation*}
\ln 1024, \quad \ln 40, \quad \ln \sqrt{\frac{2}{5}}, \quad \ln \frac{1}{10}, \quad \ln 1.024.
\end{equation*}
Expand $\left(e^x+e^{-x}\right)\left(e^y-e^{-y}\right)+\left(e^x-e^{-x}\right)\left(e^y+e^{-y}\right)$.
Expand $\left(e^x+e^{-x}\right)\left(e^y+e^{-y}\right)+\left(e^x-e^{-x}\right)\left(e^y-e^{-y}\right)$.
Solve $2^x=3$. Solve $0.5^x=3$. Solve $4^x=3$.
For which values of $x$ (if any) does $1^x=1$? For which values of $x$ (if any) does $1^x=3$?
For what values of $b$ (if any) does $0^b=0$? For what values of $a$ (if any) does $a^0=0$?
Given $\log_{10}(\log_{10}x)=6$, how many zeros are there at the end of the number $x$?
Solve $e^{x}+e^{-x}=4$.
How many solutions are there to $e^x+e^{-x}=c$? Identify different cases in terms of $c$.
Prove that $\ln(N+\sqrt{N^2-1})=-\ln(N-\sqrt{N^2-1})$ for any number $N\geq 1$.
Consider the equation $x^y=y^x$ with $x,y>0$. Use logarithms to turn this into an equation of the form $f(x)=f(y)$.
[Harder] Sketch $f(x)$.
Simplify $a^{k\log_a b}$ for positive numbers $a$, $b$, $k$.
Consider the number $x=\log_a b \log_b c$. By simplifying $a^x$, show that $x=\log_a c$.
Similarly, show that $\displaystyle \log_a b = \frac{\log_c b}{\log_c a}$ for positive numbers $a$, $b$, $c$, and hence $\displaystyle \log_a b=\frac{\ln b}{\ln a}$.

MAT questions

MAT 2007 Q1I

Given that $a$ and $b$ are positive and
\begin{equation*}
4\left( \log _{10}a\right) ^{2}+\left( \log _{10}b\right) ^{2}=1,
\end{equation*}
then the greatest possible value of $a$ is
$$\text{(a)} \quad \frac{1}{10},\qquad \text{(b)} \quad 1,\qquad \text{(c)} \quad \sqrt{10},\qquad \text{(d)} \quad 10^{\sqrt{2}}.$$

MAT 2008 Q1B

Which is the smallest of these values?
$$ \text{(a)} \quad \log _{10}\pi ,\qquad \text{(b)} \quad \sqrt{\log _{10}\left(\pi ^{2}\right) },\qquad \text{(c)} \quad \left( \frac{1}{\log _{10}\pi }\right)^{3},\qquad \text{(d)} \quad \frac{1}{\log _{10}\sqrt{\pi }} .$$

MAT 2008 Q1E

The highest power of $x$ in
\begin{equation*}
\left\{ \left[ \left( 2x^{6}+7\right) ^{3}+\left( 3x^{8}-12\right) ^{4}
\right] ^{5}+\left[ \left( 3x^{5}-12x^{2}\right) ^{5}+\left( x^{7}+6\right)
^{4}\right] ^{6}\right\} ^{3}
\end{equation*}
is
$$ \text{(a)} \quad x^{424},\qquad \text{(b)} \quad x^{450},\qquad \text{(c)} \quad x^{500},\qquad \text{(d)} \quad x^{504}. $$

MAT 2010 Q1E

Which is the largest of the following four numbers?
\begin{equation*}
\text{(a)}\quad \log _{2}3,\qquad \text{(b)}\quad \log _{4}8,\qquad \text{(c)
}\quad \log _{3}2,\qquad \text{(d)}\quad \log _{5}10.
\end{equation*}

MAT 2012 Q1C (modified)

Which is the smallest of the following numbers?
$$\text{(a)} \quad \left(\sqrt{3}\right) ^{3},\qquad \text{(b)} \quad \log _{3}\left( 9^{2}\right),\qquad \text{(c)} \quad \left( 3\sin 60^\circ\right)^{2},\qquad \text{(d)} \quad \log _{2}\left( \log _{2}\left(8^{5}\right) \right).$$

Hints

MAT 2007 Q1I

If I squint at the left-hand side, it looks a bit like the sum of two squares. Let's write $x=\log_{10}a$ and $y=\log_{10} b$ and see what happens.

MAT 2008 Q1B

When is $x$ bigger than $\sqrt{2x}$? When is $x$ bigger than $2/x$?

You'll need to use the fact that $1<\pi<10$, but you shouldn't need to use any more detailed knowledge of the value of $\pi$ than that.

MAT 2008 Q1E

What's the highest power of $x$ in $(2x^6+7)^3$? Do not multiply out! Now look at the other terms too.

We can ignore the outer-most power of 3 while we're comparing terms, but don't forget about it at the end.

MAT 2010 Q1E

You can evaluate one of these exactly. Which one? Next, I would aim to compare the others to that one.

Here's a strategy to do that sort of comparison; let's say that we're comparing $\log_2 3$ against $\frac{p}{q}$ for some fraction $\frac{p}{q}$. Is $\log_2 3 < \frac{p}{q}$? Well, if it is, then $3<2^{p/q}$, so $3^q<2^p$. You've got particular values of $p$ and $q$ in mind; go for it!

You might like to reflect on why it's OK to manipulate the inequalities like this.

MAT 2012 Q1C

Simplify each number as much as you can before doing any comparisons.

Extension

[Just for fun, not part of the MAT question]

Given a positive number $\alpha$, which is the smallest of these values? Identify the different cases according to $\alpha$.
$$\text{(a)}\quad \alpha,\qquad \text{(b)}\quad \sqrt{2\alpha}, \qquad \text{(c)}\quad \alpha^{-3} \qquad \text{(d)} \quad\frac{2}{\alpha}.$$
Which is larger, $(8!)^9$ or $(9!)^8$ ?