Question

solve for x:
X^(squarert(logx))=10^(2root2)

nicer equation in comment below!

Answer 1

\[x^\sqrt{logx}=10^{2\sqrt{2}} \]

Answer 2

@kirbykirby

Answer 3

is your log in base 10 or base \(e\) ?

Answer 4

base 10

Answer 5

Well regardless... you can try taking the log of both sides:
\[\log\left( x^{\sqrt{\log x}}\right)=\log \left( 10^{2\sqrt{2}}\right)\\
\sqrt{\log x}\cdot \log x=2\sqrt{2}\log 10, ~~\text{ using the rule} \log a^b=b \log a\\
\left( \log x\right)^{1/2}\cdot \log x=2\sqrt{2} \log 10 \\
\left(\log x\right)^{3/2}=2\sqrt{2}\log 10, ~~\text{ using the law of exponents } a^xa^y=a^{x+y}\\
\left[\left(\log x\right)^{3/2}\right]^{2/3}=\left[2\sqrt{2}\log 10 \right]^{2/3}\\
\log x = \left[2\sqrt{2}\log 10 \right]^{2/3}\]

Now here's the part where the base of the log matters... if it's base 10: then by the definition of the log, you get:
\[ x=10^{\left[2\sqrt{2}\log 10 \right]^{2/3}}=10^2=100\]
If it's in base \(e\), you get :
\[ x=e^{\left[2\sqrt{2}\log 10 \right]^{2/3}}\approx 32.7022\]

Answer 6

Ah ok lol sorry I didn't see your post above^ ignore the base \(e\) part :)

Answer 7

in fact since it's in base 10, you can simplify from the beginning that \(\log 10 =1\) to simplify things a little bit

Answer 8

lol the answer is just 10^2. i think you are supposed to do this without a gdc?

Answer 9

gdc?

Answer 10

and \(10^2=100\) as I showed above

Answer 11

ah you mean graphical display calculator? (Googled it lol)

Answer 12

oh yes!! haha sorry i didnt see that although i magnified the math notation to like 200 (im blind D:)

Answer 13

ah lol sorry ya it's a bit small with all the exponents.. hm but to simplify it in more detail:

Answer 14

why couldnt you simplify the log10 to 1? since it is base 10

Answer 15

u just use exponent rules..\[x=\large 10^{\left[2\sqrt{2}\log 10 \right]^{2/3}}=\large 10^{\left[2\sqrt{2}\right]^{2/3}}=\large10^{2^{(2/3)}2^{1/2*2/3}}=10^{2^{(2/3)+(1/3)}}10^2=100 \]

Answer 16

I just wrote it in general because I wasn't sure if it was base 10 or base e (I started writing before you replied xD), but I mentioned in my comment afterward, that you could have simplified it from the beginning since it was in base 10

Answer 17

yep i saw that! thank you so much!!!