How can you solve $5^{x^2+8}=125^{2x}$ for x?

[karush]

$\tiny [6.3.74 Miliani HS$
Find x
$5^{x^2+8}=125^{2x}$
$\begin{array}{rlll}
\textsf{common base}&125^{2x}=(5^3)^{2x}=5^{6x}\\
\textsf{then } &x^2+8=6x\implies x^2-6x+8=0 \\
\textsf{factor}&(x-2)(x-4)=0\\
\textsf{get zeros}&x=2, \quad x=4\\
\end{array}$

should be ok
suggestions...

 

[skeeter]

that’s fine

 

[HOI]

I would suggest that you CHECK your answers!

If x= 2 then $x^2+ 8= 4+ 8= 12$ so $5^{x^2+ 8}= 5^{12}= 244140625$ while $125^{2x}= 125^{4}= 244140625$. Yes, they are equal!

If x= 4
then $x^2+ 8= 16+ 8= 24$ so $5^{x^2+ 8}= 5^{24}= 59604644775390625$ while $125^{2x}= 125^{8}= 59604644775390625$. Yes, they are equal!

 

[HOI]

If you live in Hawaii, how can you bear to do mathematics rather than being out on the beach every day?

 

[karush]

there aways seems to be some non textbook trick with logs

 

[jonah1]

Beer soaked ramblings follow.

If you live in Hawaii, how can you bear to do mathematics rather than being out on the beach every day?

Not everyone is fond of the sun and the beach.
Some would rather be indoors away from pesky flies and mosquitoes.

 

[karush]

Beer soaked ramblings follow.

Not everyone is fond of the sun and the beach.
Some would rather be indoors away from pesky flies and mosquitoes.

im 76 and over weight I am embarrased to be seen in swimsuit
besides the mask restrictions have been ridiculus here. but they are getting ignored more and more finally,,
now they wondering if we will have condo colapse like florida

 

[karush]

I would suggest that you CHECK your answers!

If x= 2 then $x^2+ 8= 4+ 8= 12$ so $5^{x^2+ 8}= 5^{12}= 244140625$ while $125^{2x}= 125^{4}= 244140625$. Yes, they are equal!

If x= 4
then $x^2+ 8= 16+ 8= 24$ so $5^{x^2+ 8}= 5^{24}= 59604644775390625$ while $125^{2x}= 125^{8}= 59604644775390625$. Yes, they are equal!

actually I am more interested in the steps
probably don't need the decimal unless there is some purpose for it
I ussually check with W|A if the book does not give answers

 

[topsquark]

actually I am more interested in the steps
probably don't need the decimal unless there is some purpose for it
I ussually check with W|A if the book does not give answers

Never trust the calculator!

-Dan