How do you find out the exponent if you can't use logs?

[priceofcarrot]

There isn't a problem here, it's an example they gave, but I don't get how you can get the answer without using logs.

2^(2x+1) = 1/32
2^(2x+1) = 2^(-5)

How do they know that it's -5?


Same thing for:

3^(x-1) = 27
3^(x-1) = 3^3



Is it just trial and error? I've tried a bunch of different methods but none have worked.

 

[Mark44]

There isn't a problem here, it's an example they gave, but I don't get how you can get the answer without using logs.

2^(2x+1) = 1/32
2^(2x+1) = 2^(-5)

How do they know that it's -5?

Because 32 = 2⁵, that's how. So 1/32 = 1/(2⁵) = 2^-5.

Same thing for:

3^(x-1) = 27
3^(x-1) = 3^3

You know that 3² = 9, right, so 3³ = 27. That's all it is.

Is it just trial and error? I've tried a bunch of different methods but none have worked.

No, it's not trial and error. It's useful to know a few powers of small numbers.

2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 correspond to exponents on 2 of 1, 2, 3, ..., 10

3, 9, 27, 81, 243 correspond to exponents on 3 of 1, 2, 3, 4, and 5

and so on.

 

[priceofcarrot]

Is there any simple formula I can use to find that 32 = 2^5? Or similar things, or is it just having an idea which numbers as the exponent would produce which numbers?

Can I just knowing 32 = 2^x find out what x is with a formula?

 

[eumyang]

Is there any simple formula I can use to find that 32 = 2^5? Or similar things, or is it just having an idea which numbers as the exponent would produce which numbers?

Can I just knowing 32 = 2^x find out what x is with a formula?

Just memorize some of the powers, as Mark44 said. It really isn't much different than memorizing addition and multiplication facts (except that the numbers are bigger, of course).

 

[Mark44]

Or you might notice that 32 = 2 * 16 = 2 * 2 * 8 = 2 * 2 * 2 * 4 = 2 * 2 * 2 * 2 * 2 = 2⁵.

Do you know how to factor a number into its prime factors?

 

[Chestermiller]

Is there any simple formula I can use to find that 32 = 2^5? Or similar things, or is it just having an idea which numbers as the exponent would produce which numbers?

Can I just knowing 32 = 2^x find out what x is with a formula?

Sure. You start out with 1, and then multiply it by 2 to get 2. Then you multiply the 2 by 2 to get 4. Then you multiply the 4 by 2 to get eight. Then you multiply the 8 by 2 to get 16. Then you multiply the 16 by 2 to get 32. How many multiplications did you do all together to get 32?