# If $3^x = 5$, $5^y = 10$, $10^z = 16$, then what is $3^{xyz}$?

Can't post images so I'll type it here:

$$3^x = 5,\qquad 5^y = 10,\qquad 10^z = 16$$

Then what is $$3^{xyz}$$?

I've spent like an hour trying to solve it and I failed. Help would be super duper appreciated. Thank you!

Edit: uhh I think I solved it? Would the answer be $$16$$?

Basically I put $$3^x$$ in place of the $$5$$ in $$5^y = 10$$, so now I have $$(3^x)^y = 10$$ (which is $$3^{xy} = 10$$), did the same for the last equation and I got $$16$$ as an answer, but can anyone confirm this?

• To be clear: Do you intend $xyz$ to be the exponent on $3$? – Blue Dec 17 '18 at 23:14
• Can you give a better title? That's what everyone is doing. – stuart stevenson Dec 17 '18 at 23:14
• Hint $(p^n)^m=p^{nm}$ – randomgirl Dec 17 '18 at 23:15
• Yes the exponent on 3, and I'm sorry this is my first time here :D ALSO YES RANDOMGIRL I think I just solved it using that – Naji Nazzal Dec 17 '18 at 23:17
• Hint $\large\ 3^{\large xyz} = ((3^{\large x})^{\large y})^{\large z}\ \$ – Bill Dubuque Dec 25 '18 at 22:00

$$3^{xyz}$$ is the same as $$(3^x)^{yz}$$ and $$3^x=5$$

this becomes $$5^{yz}$$ and this is the same as $$(5^y)^{z}$$ and if $$5^y=10$$

this becomes $$10^z$$ and since $$10^z=16$$ you have that $$3^{xyz}=16$$

• That's indeed the solution, I just did not expect it to be this short of an answer. Thank you! – Naji Nazzal Dec 17 '18 at 23:30
• No problem, could you verify this as your chosen answer please, Thanks – ricky Dec 17 '18 at 23:53
• Took me a while to figure out how lmao, but I gotcha. Thanks again. – Naji Nazzal Dec 17 '18 at 23:58

Ricky gives the best solution but here is a brute force solution that uses more machinery. We note that $$x=\frac{\log 5}{\log 3};\quad y=\frac{\log 10}{\log 5}; \quad z=\frac{\log 16}{\log 10}$$ so $$xyz=\frac{\log 16}{\log 3}=\log_{3}16$$ whence $$3^{xyz}=16.$$

• That was an interesting read, thanks! I'm still not too experienced in logarithms – Naji Nazzal Dec 18 '18 at 0:00