# 6.3.74 5^{x^2+8}=125^{2x}

• MHB
• karush
In summary, To solve $5^{x^2+8}=125^{2x}$, we first use the fact that they have a common base of 5. Then, we can rewrite $125^{2x}$ as $(5^3)^{2x}$, which equals $5^{6x}$. Next, we equate the exponents and get $x^2+8=6x$. This can be factored to $(x-2)(x-4)=0$, giving us the solutions $x=2$ and $x=4$. Finally, checking both values shows that they are indeed correct.f

#### karush

Gold Member
MHB
$\tiny [6.3.74 Miliani HS$
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...

that’s fine

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!

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

there aways seems to be some non textbook trick with logs

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.

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

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

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