Question

Solve the exponential equation. Express the solution in terms of natural logarithms. Then use a calc. to obtain a decimal approximation for the solution. 4^(x+)= 301

Answer 1

x plus what?

Answer 2

oh sorry it's x+3 :)

Answer 3

@AnonymousBeast do you think you can still maybe help me out? It would be greatly appreciated :)

Answer 4

@D3xt3R Could you please help me?

Answer 5

Yes, I'm here.

Answer 6

Rewrite in logarithmic form.
Use this method:
\[a^b=y \rightarrow \log_{a}y=b \]

Answer 7

\[4^{x+3}=301\]we can write\[4^{x+3}=2^{2*(x+3)}=2^{2x+6}\]now writing with the new form\[2^{2x+6}=301\]apply natural logarithms on both sides\[ln(2^{2x+6})=ln(301)\]\[(2x+6)*ln(2)=ln(301)\]\[2x+6=\frac{ln(301)}{ln(2)}\]\[2x+6=log_{_2}(301)\]\[2x=log_{_2}(301)-6\]\[\boxed{\boxed{x=\frac{log_{_2}(301)-6}{2}}}\]

Answer 8

Or that works too :P

Answer 9

This stuff is so confusing!! Do you think you can explain it a little more maybe??

Answer 10

I can show you another way perhaps?

Answer 11

I just apply a logarithm property, and it is.\[log_c(a^b)\Leftrightarrow b*log_c(a)\]

Answer 12

So what exactly do I need to do? I am soo sorry guys! :/

Answer 13

After converting to logarithmic form, use the change of base rule:
\[\log_{4} 301=\frac{ \log 301 }{ \log 4 }\]
Use a calculator. It is of base 10, so just about all scientific calculators can solve.
You get about 4.1168098383798510231630554788499
Now we have:
4.1168098383798510231630554788499=x+3
Simplify:
x= approximately 1.1168098383798510231630554788499

Answer 14

Oh, my bad. While x is correct, I did not see to express it in NATURAL logarithms. Sorry :/

Answer 15

This is Algebra II, right?

Answer 16

It's okay :) How do you express it in natural logarithms?

Answer 17

D3xter expressed it in terms of ln. His result should be your answer I believe.

Answer 18

\[natural~logarithms~=~ln(anything)~=~log_{_e}(anything)\]

Answer 19

So, I am so confused. It's not your answer @AnonymousBeast

Answer 20

What are your choices?

Answer 21

We don't have choices :/

Answer 22

If you substitute about 1.1168098383798510231630554788499 for x into the original problem, it'll get to about 100.99... which is about as close as you'll get. Like I said, I can't express this in natural logarithms. @D3xt3R may be able to help you.

Answer 23

Guy's I'm from Brazil, can you tell me if you can use calculator??

Answer 24

I think you can

Answer 25

Can you please express what AnonymousBeast said into natural logarithms??

Answer 26

You would have to in order to do logarithms, really.

Answer 27

Do I express 100.99 in natural log?

Answer 28

@CookieMonsta22 first step, choose witch log do you wanna you, second step apply this log in both sides, third step use the properties of logarithms, then you will find the value of X, like i did, now using a calculator you have to write this sentence to find the value\[x=\frac{ln(301)}{ln(2)}-6\]you'll find\[\boxed{\boxed{x\approx2.2336}}\]

Answer 29

So is x=2.2336 the final answer?

Answer 30

Yes

Answer 31

Oh my gosh thank you so so so much!!! Both of you are amazing! I cannot thank you enough :) <3

Answer 32

How do I award you guys medals??

Answer 33

@AnonymousBeast and @D3xt3R how do I award you medals?

Answer 34

Do you guys know what the solution in terms of natural logarithms is? It asks for the solution set??

Answer 35

The solution of a natural logarithms is a rational number or an irrational number

Answer 36

@CookieMonsta22 I read you question again, and the right answer is \[x=\frac{ln(301)-6~*~ln(2)}{ln(2)}\]you have to do the minimum common multiple (MCM), and don't have to disappear with the natural logarthms.

Answer 37

Is that the solution set @D3xt3R

Answer 38

Yes, you don't have to replace the real value of ln(301) and ln(2), you just have to answer like I did.

Answer 39

Okay so that's the final answer for the solution set?

Answer 40

@D3xt3R so that's the final answer for the solution set?