Question

8^x=16

solve for x.

Answer 1

I just made the dumbest mistake ever. Studying too hard X) One sec

What's a common base for 8 and 16?

Answer 2

8

Answer 3

We see that we can raise 2 to a certain power to get both 8 and 16
\(2^3=8\)
\(2^4=16\)

Answer 4

Using this knowledge, we can make proper substitutions:
\[\huge 2^{3x}=2^4\]

Answer 5

We need to solve for x. But to make this a true statement, we have to solve 3x=4

Answer 6

Once we've reached a common base, we don't need to worry about it anymore

Answer 7

So what is x if we solve 3x=4?

Answer 8

are we multiplying 3 and x to get 4?

Answer 9

Yes, but what times 3 is equal to 4?

Answer 10

1.333333 equals 3.999999 but you can't get 4 exactly

Answer 11

What? This is a simple algebra task. You were right with the 1.333333 which in fractional form is 4/3. Not sure what you're doing with the rest of it here

Answer 12

So \(x=4/3\)

That is the solution to the question. And you will see that if you enter into a calculator\[8^{4/3}\]Then you'll get 16

Answer 13

Oh I get it now 8^4/3=16! I was overthinking haha

Answer 14

I skipped a step earlier, but really what we're doing is this\[\huge 8^x=(2^3)^x=2^{3x}\]

Answer 15

Yep!! X)

Answer 16

Thank you so much! Is there anyway you could help me with another?

Answer 17

I can try really quickly X) I have to go and study for finals soon :|

Answer 18

7^(-x+3)=46

Answer 19

In the previous problem we could make that simple process because we matched the bases. But here we can't do that.

I would use logs/natural logs. Have you learned about them yet?

Answer 20

eg: \(\log(x) ~~~\text{and}~~~\ln(x)\)

Answer 21

i dont get how to use them in problems like this

Answer 22

Sure, it's actually pretty simple.

There's a property of logs/natural logs that says this:\[\huge \ln(a^x) \implies x \ln(a)\]

Therefore, we can take the natural log of both sides and get\[\ln(7^{-x+3})=\ln(46)\]\[(-x+3) \ln(7)=\ln(46)\]And continue to solve for x by dividing by ln(7). It's also best to not use a calculator and evaluate as a decimal and instead just leave it in logarithmic form.

Answer 23

so -x+3)ln(7)=9.493682849
ln(46)=3.828641396
is that right?

Answer 24

I said don't evaluate it as a decimal X) Leave it in it's log form.

Answer 25

\[-x+3=\frac{\ln(46)}{\ln(7)}\]and solve for x from there

Answer 26

My class wants the decimals...

Answer 27

1.032467533!

Answer 28

Ahh, I see. That's kind of ridiculous in my opinion. Even to simply grade, this would be much cleaner. But oh well. Solve for x and continue to keep it in log form and then plug it into a calculator.

Answer 29

Yeah, that's what I got

Answer 30

yay! thank you so much!

Answer 31

You're welcome X)