Question

Solve the following equation:
3^{x}*5^{x-2} = 3^{4x}

I would like a hint, not a solution.

Answer 1

1) divide both sides by
\[3^x\]
2) take the log of both sides to get the exponents on the ground floor instead of in the sky
3) do a bunch of algebra

Answer 2

How about this strategy:
Take the log base 3 of both sides

Answer 3

log[3]{3^x * 5^{x-2}} = log[3]{3^{4x}}

Answer 4

Using the product rule for Logs, as well as the power rule, we get:

Answer 5

log[3]{3^x} + log[3]{5^{x-2}} = 4x*log[3]{3}

Answer 6

x + (x-2)*log[3]{5} = 4x

Answer 7

x + x*log[3]{5} - 2log[3]{5} = 4x

Answer 8

Ummm

Answer 9

Is this wrong?

Answer 10

I want to know if the method I took above would eventually work out.

Answer 11

start with
\[5^{x-2}=3^{3x}\]

Answer 12

Using the approach I took, I get x = ( 2*log[3]{5} / log[3]{5} - 3 )

Answer 13

then take the log get
\[(x-2)\ln(5)=3x\ln(3)\]

Answer 14

WOW!

Answer 15

then multiply out to get
\[\ln(5)x-2\ln(5)=3\ln(3)x\]

Answer 16

I just saw that now; the quotient rule for exponents used on 3^4x and 3^x

Answer 17

OHHH That makes it so much simpler

Answer 18

so \ln(5)x-3\ln(3)x=2\ln(5)\]
\[(\ln(5)-3\ln(3))x=2\ln(5)\] and last step is to divide.

Answer 19

oops
\[\ln(5)x-3\ln(3)x=2\ln(5)\]

Answer 20

then
\[(\ln(5)-3\ln(3))x=2\ln(5)\]

Answer 21

How do you type like that?

Answer 22

and finally
\[x=\frac{2\ln(5)}{ln(5)-3\ln(3)}\]

Answer 23

latex. i am typing latex commands

Answer 24

But aren't I using that too? I am using it in this normal box.

Answer 25

actually that is the reason i started answering to practice my latex. now i am hooked. i am writing them by hand. like rolling your own

Answer 26

start with \[

Answer 27

end with
\]

Answer 28

\[test \]

Answer 29

Thank you.

Answer 30

and put your equations in between
try \frac{a}{b}