Which is the largest of the following four numbers?

  1. \(\log_2 3\),

  2. \(\log_4 8\),

  3. \(\log_3 2\),

  4. \(\log_5 10\).

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.