# Weird logarithmic equations

## Homework Statement

I got this problem in an exam today, and I think I may have the right answer but I don't understand how I'm supposed to know if I do. It's very confusing.

## Homework Equations

If logb (9) = 1.5 and logb (2) = 0.5 what is logb (5throot(36))?

## The Attempt at a Solution

So here it seemed futile to try and find b because it seems to have 2 different values at once. Here's what I did...:

logb 9 + logb 2 + logb 2 = 1.5 + 0.5 + 0.5
logb 36 = 2.5
1/5 logb 36 = 0.5
logb 5throot(36) = 0.5

I am confused on your notation. What base are we working in? b or ()

It is base b.

HallsofIvy
Homework Helper
What you've done is completely correct. If $log_b 9= 1.5$ and $log_b= .05$, then $log_b ^5\sqrt{36}= 0.5$. But this is a very peculiar question- you are also correct that there is no value of b satisfying those conditions. In fact, $log_b 2= 0.5$ and $log_b ^5\sqrt{36}= 0.5$ is obviously impossible since $2 \ne ^5\sqrt{36}$!

since $2 \ne ^5\sqrt{36}$!

It does to one decimal place. The question is testing the students' knowledge of the rules of logarithms. The IF in the question does give some leeway.

Essnov is obviously smart enough to realise that log_b(9)=1.5 and log_b(2)=0.5 implies two different values for b, which confused him (her?)

So in trying to keep the question simple (introducing an explanation about decimal places into the question would just misdirect a lot of students) it has placed concerns into Essnovs mind.

Well done Essnov for answering it as it was wanted despite this.

Last edited:
AlephZero
Homework Helper
Maths is quite hard enough when it is written down precisely.

Pretending that 2^5 = 36, not 32 isn't maths, it's nonsense.

If log_b(9)=1.5 how do you write b precisely?

cristo
Staff Emeritus
$$log_b9=\frac{ln9}{lnb}=1.5 \Rightarrow lnb=\frac{ln9}{1.5} \Rightarrow b=e^{ln9/1.5}$$

AlephZero
Homework Helper
$$b = 81^{1/3}$$ would be another way.

I agree. Now suppose this was about learning to use a mathematical technique in say an engineering context and hence you needed a decimal number so you could apply a measurement. How precise would you give the answer?

cristo
Staff Emeritus
I agree. Now suppose this was about learning to use a mathematical technique in say an engineering context and hence you needed a decimal number so you could apply a measurement. How precise would you give the answer?

In this case, I would use more than one decimal place to avoid the confusion that has arisen! You say

Essnov is obviously smart enough to realise that log_b(9)=1.5 and log_b(2)=0.5 implies two different values for b, which confused him (her?)

Now, if I was teaching, I would set the question such that the "smart enough" students are not punished for being intelligent enough to find the question confusing.

But then, I'm not a teacher. However, there are plenty of questions one can ask to enable the student to practice the rules of logarithms, without causing such confusion!

AlephZero
Homework Helper
A better version of the question might be

If logb (9) = x and logb (2) = y, what is logb (5throot(36))?

That question requires the same knowledge of log functions to solve it, but skips the arithmetic.

A better version of the question might be

If logb (9) = x and logb (2) = y, what is logb (5throot(36))?

That question requires the same knowledge of log functions to solve it, but skips the arithmetic.

But using algebra adds an extra level of difficulty.

[as an aside in this case I would state:

When logb (9) = x and logb (2) = y, what is logb (5throot(36))?

]

Would you settle for

If logb(9) = 1.51 and logb(2) = 0.48, what is logb(5throot(36))? ?

Thanks all for input.

Got my exam results yesterday - scored 100%.