# Logarithms

Part of the Oxford MAT Livestream.

## 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 $(9!)^8$ ?