Question

log question

Answer 1

\[7^{x+2}=e ^{17x}\]

Answer 2

I got the first part\[7^{x+2}=24154952.15^{x}\]

Answer 3

lol dat first step :)
That gives us an approximation.
Let's stick with exact values for right now.

I think a good first step is to take the natural log of each side.

Answer 4

\[\Large\rm \ln(7^{x+2})=\ln(e^{17x})\]Applying our log rule gives us,\[\Large\rm (x+2)\ln(7)=17xln(e)\]Understand that step? :o

Answer 5

Yep. can you use log there? It says simplify using logs.

Answer 6

Well you can use any "log" you want,
I chose natural log because it will give us the cleanest looking answer.
If that's what you were asking...

From that point, we would distribute the log7 to each term in the brackets.

Answer 7

\[\Large\rm (x+2)\color{orangered}{\ln(7)}=17xln(e)\]
\[\Large\rm x\color{orangered}{\ln(7)}+2\color{orangered}{\ln(7)}=17xln(e)\]Yah?
Following?
Or I confuse you from the start?

Answer 8

yep, makes sense so far.

Answer 9

How bout that ln(e), what can we do about that? :d

Answer 10

ln(e)=1
correct?

Answer 11

Ok good good.\[\Large\rm (\ln7)x+2(\ln7)=17x\]

Answer 12

Don't let the log confuse you, it's just a NUMBER.
Think of it like this:\[\Large\rm ax+b=cx\]That's the form that we currently have.
So how are we going to solve for x? :d

Answer 13

hmm. divide both sides by (ln7)x?

Answer 14

ha no subtract, not divide.

Answer 15

Yah subtract seems better :d\[\Large\rm 2(\ln7)=17x-(\ln7)x\]

Answer 16

Then, some of that factoring business maybe?

Answer 17

You'd have\[x(17-\ln7)\] on the left side

Answer 18

\[\Large\rm 2\ln7=x(17-\ln7)\]Bam, there we go.
Cool, one last step to go.

Answer 19

divide.\[\frac{ 2\ln7 }{ 17-\ln7 }=x\]

Answer 20

Yay, good job.

Answer 21

could you go 2/17=x?

Answer 22

Wrastled up a victory.

Answer 23

Noooo :O
You'd have to divide each term in the denominator by 2, not just the left one.

Answer 24

ohh, ok. i see it now.

Answer 25

If you prefer a numerical answer, you could throw it into a calculator:\[\Large\rm (2\ln(7))\div(17-\ln(7))\]Something like that.

Answer 26

A numerical approximation I mean*

Answer 27

Ok, good deal. Thanks!