Which is the largest of these four logarithms?

Question

Which is the largest of the following four numbers?

$\log_2 3$ ,

$\log_4 8$ ,

$\log_3 2$ ,

$\log_5 10$ .

Suggestion

Can we evaluate any of the logarithms exactly?

What do we know about $\log_a b$ when $0 < b < a$ ?

Solution

Useful fact; if $0 < b < a$ then $\log_ab < 1$ , since $b < a^1$ .

Firstly, we notice that $\log_3 2 <1$ , while the other logarithms are greater than one, so (c) cannot be the answer.

Secondly, we might want to calculate some of the logarithms.

Useful fact; suppose we know logs to base $j$ and want to know logs to base $k$ .

Then if $\log_kx = y$ , we have $x = k^y$ , and taking logs to base $j, \log_jx = y\log_jk$ , so $y = \log_kx = \dfrac{\log_jx}{\log_jk}$ .

Since both $4$ and $8$ are powers of $2$ , $\log_4 8$ can be easily calculated. $\log_4 8= \frac{\log_2 8}{\log_2 4}= \frac{3}{2}$

At this point we will compare $\log_2 3$ and $\log_5 10$ with $\frac{3}{2}$ .

The $\iff$ symbol is really useful. $A \iff B$ means that $A$ is true if, and only if, $B$ is true. Another way to say this is that $A$ implies $B$ , and $B$ implies $A$ ; $A$ and $B$ are always true together, or false together.

We start by supposing $\log_2 3> \frac{3}{2}$ and follow through the logical implications to see if it is true. $\log_2 3> \frac{3}{2} \iff 2\log_2 3>3 \iff \log_2 9>3 \iff 9>2^3 \text{, which is true.}$ Therefore, $\log_2 3>\log_4 8$ . Similarly, $\log_5 10< \frac{3}{2} \iff \log_5 100 < 3\iff 100<5^3\text{, which is true.}$ Hence, $\log_5 10<\log_4 8$ . From the results above, we can conclude that the biggest number is $\log_2 3$ , so the answer is (a).

Alternatively, we could argue we know the biggest is either $\log_23=x$ , or $\log_510 = y$ .

So $2^x =3$ , and since $2\sqrt2 = 2(1.414...) < 3$ , $x > 1.5$ .

But $5^y = 10$ , so $5^{y-1} = 2$ , and since $\sqrt5 > 2, y-1 < 0.5$ , so $y < 1.5$ . So $x$ is the biggest.