Question

ln(x)/ln(1.5) - ln(4)/ln(1.5) = 0.5 solve for x

Answer 1

\[\Large \frac{\ln x}{\ln1.5}-\frac{\ln4}{\ln1.5}\quad=\quad 0.5\]Hmm I guess we could start by multiplying both sides by ln1.5.\[\Large \ln x- \ln 4 \quad=\quad 0.5\cdot \ln1.5\]Ok with that step?

Answer 2

Yes

Answer 3

So next we'll need to apply a few rules of logs, this might seem a little tricky so let's do them one at a time.

On the left side, we'll want to apply this rule:\[\Large \color{teal}{\log(a)-\log(b)\quad=\quad \log\left(\frac{a}{b}\right)}\]

Answer 4

Applying this rule, what will that give us on the left side of our equation? :)

Answer 5

Log (x)-log (4)

Answer 6

No no :)

`log(x)-log(4)`
and
`ln(x)-ln(4)`
are the same thing in this case.
Ignore that part of it, what does it change to (the right side) when we apply the rule?

Maybe I should have written the rule like this:\[\Large \color{teal}{\ln a-\ln b\quad=\quad \ln\left(\frac{a}{b}\right)}\]

Answer 7

I guess I dont quite get what to put in for a and b

Answer 8

It's just a rule to follow D:

So in our case we have a=x, b=4
So it will simplify to,\[\Large \ln x-\ln 4 \quad=\quad \ln\left(\frac{x}{4}\right)\]See how that follows the format of the rule in blue? :o

Answer 9

Oh ok

Answer 10

So here's what we've got so far:\[\Large \ln\left(\frac{x}{4}\right) \quad=\quad 0.5\cdot \ln1.5\]

Answer 11

That we do :)

Answer 12

Oh ok

Answer 13

I guess I dont quite get what to put in for a and b

Answer 14

Log (x)-log (4)

Answer 15

Yes

Answer 16

So our next rule we want to apply:\[\LARGE \color{royalblue}{b\cdot \ln(a)\quad=\quad \ln(a^b)}\]We're going to apply this to the right side of our equation.

Uh oh the chat is gettin all jumbled up :( weird..

Answer 17

Why is that the next rule?

Answer 18

The step we apply `after` this one will get rid of the natural logs, but not if we have a coefficient in front of the log.

That 0.5 is causing a problem, so we need a way to bring it inside of the log before we can proceed.

Answer 19

Oh ok

Answer 20

Ln(0.5^1.5) ?

Answer 21

Woops! You have the numbers backwards there.

Answer 22

Ohh whoops

Answer 23

Fixed it

Answer 24

So we get this on the right side, yes?
Ln(1.5^0.5)

Answer 25

So that brings us to this point:\[\Large \ln\left(\frac{x}{4}\right) \quad=\quad \ln\left(1.5^{0.5}\right)\]

Answer 26

When we raise something to the 0.5 or 1/2 power, that's the same as taking the square root.

So we can clean up this expression a tad by rewriting it as a square root.\[\Large \ln\left(\frac{x}{4}\right) \quad=\quad \ln \sqrt{1.5}\]

Answer 27

The next step would be to notice that both sides have the same operation being applied to them (the natural log), so the contents of those logs will be equal.\[\Large \frac{x}{4}\quad=\quad \sqrt{1.5}\]

Answer 28

Ln cancel right? Final answer 4.898979486?

Answer 29

Mmm yah that sounds right! :)

Answer 30

Awesome! Thanks so much!