Question

Solve the equation 2(3^x) = 3(4^x). Give an exact answer and a decimal approximation rounded to one decimal place.

Answer 1

log(2) + log(3^x) = log(3) + log(4^x)
= log(2) + x log(3) = log(3) + x log(4)
= x log(3) - x log(4) = log(3) - log(2)
= x(log(3)-log(4) / log(3) - log(4) = log(3) - log(2) / log(3) - log(4)

The answer should be approximately 1.4 but I can't get the right answer

Answer 2

I get -.7558684902 on my calculator

Answer 3

Did I make an error in one of my steps?

Answer 4

Not sure of the question.

Answer 5

Well, I have to give an exact answer and a decimal approximation but I'm not sure I get two answers.

Answer 6

My teacher has the answer as approximately - 1.4

Answer 7

\[\frac{2}{3}\times3^{x}=4^{x}\]
\[\ln \frac{2}{3}+x \ln 3=x \ln 4\]
-0.4005 + 1.0986x = 1.3863x
0.2877x = -0.4055
x = -1.41

Answer 8

so was I going about it the wrong way with all my log steps?

Answer 9

Logs are needed, but just in one step.

Answer 10

How did you get 2/3 instead of log(2)?

Answer 11

Sorry, I missed out my first step. This was to divide both sides of the original equation by 3.

Answer 12

2(3^x) / 3 = 3(4^x) / 3

Answer 13

2(3^x) / 3 = 3(4^x) / 3
Which can be written as:
\[\frac{2}{3}\times3^{x}=\frac{3}{3}\times4^{x}\]

Answer 14

OK, I got a different answer when I took the log of 2/3

Answer 15

ln(2/3) = -.4055 + 1.0986 = 1.3863
.6931x = 1.3863
1.3863 / .6931 = 2.0001

Answer 16

I did something wrong because not ending up with 1.4

Answer 17

How did you get -.4005 for ln(2/3)?

Answer 18

It is coming up on my calculator as -.4054

Answer 19

Sorry, it is bad copying on my part. I typed -0.4005 instead of -0.4055. However the rest of what I typed in my original post did not carry the error into calculation.

Answer 20

So tomorrow for my final, when the teacher wants an exact answer and a decimal approximation, that just means not to round?

Answer 21

-.4055 + 1.0986 = 1.3863 rounded to one decimal place is 1.4
.6931x = 1.3863, solve for x and that is exact answer?

Answer 22

She only gave -1.4 as the answer, so I don't know what she means by an exact answer and a decimal approximation

Answer 23

-0.4055 + 1.0986x = 1.3863x
0.2877x = -0.4055
x = -1.40945 (exact answer)
x = -1.4 (rounded to one decimal place)

Answer 24

Where does the - come from? I'm thinking -.4055 is positive

Answer 25

Only because you are adding 1.0986 which is a bigger number

Answer 26

-0.4055 + 1.0986x = 1.3863x
Do you agree so far, that -0.4055 is valid being the natural log of 2/3?

Answer 27

Yes, I agree with that

Answer 28

-0.4055 + 1.0986x = 1.3863x
Subtracting 1.0986x from both sides gives:
-0.4055 = 1.3863x - 1.0986x = 0.2877x ...........(1)
We can interchange the left and right sides of equation (1) to give:
0.2877x = -0.4055
Are you with me here?

Answer 29

OK, I'm up to that point

Answer 30

I did .2903 and not .2877

Answer 31

I did get .2903 rather than .2877

Answer 32

So do you agree that you made an error when subtracting:
1.3863x - 1.0986x

Answer 33

I agree but now to find the error...

Answer 34

Just found it, I entered it incorrectly

Answer 35

np. We can all do that :)

Answer 36

The final step is just to divide by .2877 on both sides

Answer 37

x is approximately 1.409454293

Answer 38

That should be -1.409454293

Answer 39

Looks good. Perhaps taking it to five decimal places is enough for an 'exact' answer. Actually, strictly speaking, it is not possible to give an exact answer in such a calculation.

Answer 40

I would just her an answer with all the decimal places on the calculator.

Answer 41

give

Answer 42

Hopefully that will get full marks.