Exponents (URGH!!!)

Okay, for example:

2^x = 16; x = 4 (I find this by 2*2*2*2)

Q1: Is there a better way to do this? If I had 3^x = 243, it would take a while to
get to x = 5 without a calculator on a test.

x^4 = 16; x = 2 (I find this by taking the 4th root of 16)

BUT ...

When the answer isn't obvious, how do I find the value of x? For example:

Q2: 8^x = 0.25 (I know x = -2/3, but I don't know how to get that answer)

Q3: 10^x = sqrt10 (I know x = 1/2, but why?)

Q4: 5^x = 0.2 (Again, I know x = -1, but don't understand why)

Thanks!
Angel

Angel - have you learnt logs and how to use them

8^x = 0.25

log 8 ^ x = log 0.25

x log 8 = log (1/4)

x log 2^3 = log 1- log 2^2

3x log 2 = 0 - 2 log 2

3x = -2
x=-2/3

Oh, I see! You made that so easy!

I thought I was forgetting some exponent rules or something . . . just needed to apply log properties.

Thank you so very much -- you saved me many hours tonight
Angel

Hello, Angel!

For example: .\(\displaystyle 2^x\,=\,16\;\;\Rightarrow\;\;x = 4\) . (I find this by 2*2*2*2)

Q1: Is there a better way to do this?
If I had \(\displaystyle 3^x\,=\,243\), it would take a while to get to \(\displaystyle x = 5\) without a calculator on a test.

Click to expand...

Without a calculator, that's the only way to do it . . . sorry.

\(\displaystyle x^4\,=\,16\;\;\Rightarrow\;\;x = 2\) . (I find this by taking the 4th root of 16)

BUT ... When the answer isn't obvious, how do I find the value of x?

For example:
\(\displaystyle Q2:\; 8^x\,=\,0.25\)
I know \(\displaystyle x = -\frac{2}{3}\), but I don't know how to get that answer

\(\displaystyle Q3:\;10^x\,=\,\sqrt{10}\)
I know \(\displaystyle x = \frac{1}{2}\), but why?

\(\displaystyle Q4:\;5^x\,=\,0.2\)
Again, I know \(\displaystyle x = -1\), but don't understand why

Click to expand...

Are you familiar with the "Equal bases, equal exponents" rule?

If we have: .\(\displaystyle b^x\,=\,b^y\), then \(\displaystyle x\,=\,y\).
. . Given two equal exponentials, if the bases are equal, then the exponents are equal.


\(\displaystyle Q3:\;10^x\,=\,\sqrt{10}\)
. . . . Try to get the same base on both sides.
. . . . Hey, they <u>are</u> the same . . . both 10's.
. . . . So their exponents are equal:
. . . . . . \(\displaystyle 10^x\,=\,10^{\frac{1}{2}}\;\;\Rightarrow\;\;x\,=\,\frac{1}{2}\)


\(\displaystyle Q2:\;5^x\,=\,0.2\)
. . . . Try to get the same base on both sides.
. . . . The left side has base \(\displaystyle 5\).
. . . . The right side is: \(\displaystyle 0.2\,=\,\frac{1}{5}\,=\,5^{-1}\) . . . there!
. . . . We have: .\(\displaystyle 5^x\,=\,5^{-1}\;\;\Rightarrow\;\;x\,=\,-1\)


\(\displaystyle Q1:\;8^x\,=\,0.25\)
. . . . Try to get the same base on both sides.
. . . . The left side has base \(\displaystyle 8\).
. . . . The right side is: \(\displaystyle 0.25\,=\,\frac{1}{4}\,=\,4^{-1}\) .(base 4)

Oh-oh! .We have two different bases (8 and 4).
. . Now what?

Sometimes we have to change <u>both</u> bases . . .

The left side is: .\(\displaystyle 8^x\,=\,(2^3)^x\,=\,2^{3x}\)

The right side is: .\(\displaystyle 4^{-1}\,=\,(2^2)^{-1}\,=\,2^{-2}\)

. . So we have: .\(\displaystyle 2^{3x}\,=\,2^{-2}\) . . . same base!

Equate exponents: .\(\displaystyle 3x\,=\,-2\;\;\Rightarrow\;\;\,=\,-\frac{2}{3}\) . . . There!