Review question

# Which is the largest of these four logarithms? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R7231

## Solution

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.