But we can't compare them!

Algebra Level 1

Which of the following logarithms is greater? $\color{#20A900}{\log _{9}{71}} ~~~~\text{or}~~~~\color{#3D99F6}{\log _{8}{61}}$

\log_{9 }{71}

\log_{8}{61}

3 solutions

Brian Charlesworth
Nov 3, 2015

Note first that $\dfrac{61}{64} \gt \dfrac{71}{81}.$

(Proof: We have that $\dfrac{30}{640} \lt \dfrac{30}{243} \Longrightarrow \dfrac{3}{64} \lt \dfrac{10}{81} \Longrightarrow 1 - \dfrac{3}{64} \gt 1 - \dfrac{10}{81} \Longrightarrow \dfrac{61}{64} \gt \dfrac{71}{81}.$ )

By the change of base rule we then have that

$\log_{8}\left(\dfrac{61}{64}\right) \gt \log_{8}\left(\dfrac{71}{81}\right) = \dfrac{\log_{9}\left(\dfrac{71}{81}\right)}{\log_{9}(8)} \gt \log_{9}\left(\dfrac{71}{81}\right),$

since we also know that $0 \lt \log_{9}(8) \lt 1.$ This implies that

$(\log_{8}(61) - \log_{8}(64)) \gt (\log_{9}(71) - \log_{9}(81)) \Longrightarrow \boxed{\log_{8}(61) \gt \log_{9}(71)},$

since $\log_{8}(64) = \log_{9}(81) = 2.$

Whenever I see the word "clearly" in a solution, I think the writer has not bothered to write up an explanation. The point of a solution is to work through the problem so that it makes sense to anyone! "Clearly", "evidently", "obviously" are not, in my opinion, words used in a good solution.

A Former Brilliant Member - 5 years, 7 months ago

Thanks for the scolding, Andrei. :) I do realize that the use of these words raise red flags for some, but in my opinion they can be used when writing for specific audiences, (although I do draw the line at the word "obviously"; that one is a definite non-starter in any situation). I felt that, for a level 3 problem solver, the fact that $\frac{61}{64} \gt \frac{71}{81}$ would be quite evident upon observation, hence I felt I could use the word "clearly" without appearing intimidating or condescending. However, out of respect for you and your criticism, I have edited my solution to include a brief proof of my initial claim. I hope that you find it suitable.

In considering how to phrase a solution, there is a fine line between offering too much exposition and too little. For some readers "too much" can be tedious and unnecessary, and for others "too little" can be cryptic and unsatisfying. I prefer brevity so long as it does not compromise clarity, and in this case I felt that my initial claim was sufficiently self-evident to justify the use of the word "clearly" without further explanation. That being said, I did find it an interesting exercise to provide as straightforward a proof as possible. :)

Brian Charlesworth - 5 years, 7 months ago

Hjalmar Orellana Soto
Nov 4, 2015

$61$ is closer enough to $8^{2}$ than $71$ is to $9^{2}$ , so $log_{8}61$ is greater. That was my reasoning

While that's clever intuition that worked out here, it turns out that this is not always the case. For example, 55 is closer to $8^2=64$ than 71 is to $9^2=81,$ but $\log_{8}(55) < \log_{9}(71).$

Eli Ross Staff - 5 years, 7 months ago

Yes, we even have that $\log_{8}(56) \lt \log_{9}(71),$ which would be quite risky to intuit.

Brian Charlesworth - 5 years, 7 months ago

This intuition is dependent on the ratio of given values. I am not a real analysis student, not currently, maybe in future I will do. But my mathematical experience tells me that analysing such things requires the analysis of their ratios.

Laxmi Narayan Bhandari Xth B - 8 months, 3 weeks ago

Harmony Luce
Mar 30, 2016

By change-of-base, we have

$\log_8 61 = \dfrac{\log 61}{\log 8} = \dfrac{\log 61}{3\log 2} = \dfrac{1}{3}\log_2 61$

and

$\log_9 71 = \dfrac{\log 71}{\log 9} = \dfrac{\log 71}{2\log 3} = \dfrac{1}{2}\log_3 71$ .

Note: Both $\log_2 61$ and $\log_3 71$ are positive.

Observe that,

$\log_2 61 < 6 \implies \dfrac{1}{3}\log_2 61 < 2$

and

$\log_3 71 < 4 \implies \dfrac{1}{2}\log_3 71 < 2$ .

Notice that it only takes one-third of $\log_2 61$ to be less than 2, whereas it takes one-half of $\log_3 71$ to be less than 2. Since both $\log_2 61$ and $\log_3 71$ are positive, it must follow that $\log_2 61 > \log_3 71$ .

Equivalently, $\log_8 61 > \log_9 71$ .

But we can't compare them!

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