Exponential Equations...Part 2

[greatwhiteshark]

1) 3^x = 9^x

1) Make both bases the same.

3^x = 3^2x

2) Equate exponents

x = 2x

3) Set to zero

0 = 2x - x

4) factor the left side

0 = x(2 - 1)

0 = x

Check:

3^x = 9^x

3^0 = 9^0

1 = 1

2) 2^x = 10

I don't know how to break exponential equation 2. Help.

 

[tkhunny]

x = 2x

3) Set to zero

0 = 2x - x

4) factor the left side

0 = x(2 - 1)

What? Simplify to 2x - x = x. Then what is there to factor?

2) 2^x = 10

Can you just use logarithms? The bases don't HAVE to be the same.

x*log(2) = log(10) = 1
x = 1/log(2)

It's not as pretty.

 

[greatwhiteshark]

okay

Your statement:

"What? Simplify to 2x - x = x. Then what is there to factor?"

Yes, but does x = 0 at the end of the problem?
I substituted o for x in the original equation and produced a a balance os
1 = 1. Is this right?


For question 2^x = 10, I cannot use logs because our teacher has not taught logs. He will get into logs next week. How can I solve
2^x = 10 without using logs?

 

[soroban]

Re: okay

Hello, Janet!

For question 2^x = 10, I cannot use logs
because our teacher has not taught logs.
He will get into logs next week.

Then he shouldn't have assigned that problem.

How can I solve 2^x = 10 without using logs?

You can't . . . unless you want to <u>estimate</u> the answer with a LOT of trial-and-error.

We know that: . 2³ .= .8 . . . too small
We know that: . 2⁴ .= 16 . . . too big
. . So x is between 3 and 4.

We try x = 3.3: . 2^3.3 .= .9.849...
We try x = 3.4: . 2^3.4 .= 10.556...
. . So x is between 3.30 and 3.40

We try x = 3.32: . 2^3.32 .= .9.9866...
We try x = 3.33: . 2^3.33 .= 10.056...
. . So x is between 3.320 and 3.330

. . . and so on . . . a very primitive way get the answer.

I'd wait until he teaches you logs.

 

[tkhunny]

Re: okay

Yes, but does x = 0 at the end of the problem?

The fact that one of your factors has only coefficients should have tipped you off that something was a bit odd. It leads to the same solution. I wouldn't call it "wrong" the way you did it. It's just unnatural looking.

 

[greatwhiteshark]

Hey

Thank you all for your help and great notes.