Exam problem : solve exponential withh log? This one is killin me

[sixshot8]

exam problem : solve exponential withh ... log? This one is killin me!

36^x-6*6^x=-9

please show and explain steps

 

[cepheid]

Welcome to PF sixshot8!

Remember that the logarithm to base "n" of something is the inverse operation of raising n to the power of that something i.e.:

log_n(n^x) = x

Now, you can rewrite every term on the left hand side as a power of 6:

36^x = (6²)^x = ?

6*6^x = 6¹6^x = ?

Use what you know about the rules of exponents to fill in the question marks.

So, it seems like at some point, taking a base 6 logarithm might be useful. Or you can express the exponentials in terms of another base, like 'e', that might be more convenient.

We don't do people's homework for them at PF, so I'm not going to give you a step by step solution.

EDIT: Looking at things more closely, I think it might be useful to express each term as a multiple of 6^x

 

[eumyang]

Similar to what cepheid originally said, make a substitution where w = 6^x. Rewrite the original equation in terms of w, and go on from there.

 

[AJKing]

Which equation are you asking about?

$36^{x-6}*6^x = (-9)$

or

$36^{x}-6*6^x = (-9)$

I would think you mean the first, correct?

 

[Mentallic]

Which equation are you asking about?

$36^{x-6}*6^x = (-9)$

or

$36^{x}-6*6^x = (-9)$

I would think you mean the first, correct?

The second seems more likely considering how often they like to bring up these kinds of questions.

 

[Skins]

You have

$$36^{x}-6(6^{x}) = -9$$

which you can also write as

$$36^{x}\, -\, 6(6^{x}) \, + \,9=0$$

Now

$$36^{x} \, = \, (6.6)^{x} \, = \, (6^{2})^{x} \, = \, (6^{x})^{2}$$

So your above equation can be written as

(**) $$(6^{x})^{2} \, - \, 6(6^{x}) \, + \, 9 \, = \, 0$$

Does equation (**) seem to have a familiar (common) looking form that you have seen before ? Think about it. How would you go about solving for
$$6^{x}$$ ?

 

[Redbelly98]

Moderator's note:

Now that the OP has received plenty of hints, let's wait for a response before offering further help.

 

[Borek]

Which equation are you asking about?

$36^{x-6}*6^x = (-9)$

or

$36^{x}-6*6^x = (-9)$

I would think you mean the first, correct?

Slightl;y OT, but I don't see how the first could give negative result.