Question

Solve for x using logs.

8 * 5^x = 7 * 3^x

x=________.

How do I solve this using logs? :/

Answer 1

Start by taking logs of both sides.

Answer 2

so log(8 * 5^x)= log(7*3^x) ?

or is it just log(8) * 5^x = log(7) * 3^x ?

Answer 3

the first option

then split it up like the previous problem

remember
\[\ln a^x = x \ln a\]

Answer 4

oh okay :)

sorry what does that last part say? latex still isn't showing up :(

Answer 5

oh
log(a^x) = xlog(a)

Answer 6

so log(8) x log(5^x)=log(7) x log (3^x)

is that how you split it up?

Answer 7

log(ab) = loga + logb
so
log8 + log 5^x = log7 + log3^x

Answer 8

ohh okay so would it become this?

5^x log (8) = 3^x log (7) ?

Answer 9

oh so i don't need the parentheses?

Answer 10

No,

log(8 * 5^x)= log(7*3^x)
becomes
log8 + log 5^x = log7 + log3^x

Answer 11

okay

so it simplifies to this?

.903 + log 5^x = .845 + log 3^x

?? not really sure how to simplify the logs with the x as the exponent though :/

Answer 12

use the property i posted ^^^

Answer 13

okay

so log(a^x) = xlog(a)

that makes it

.903 + xlog(5) = .845 + xlog(3) ?

Answer 14

:)

Answer 15

.903 + x(.699) = .845 + x(.477)

is that right? :/

Answer 16

yeah, by changing the logs to rounded decimals it may effect the precision of your final answer
just a warning

Answer 17

ohh okay here are the exact numbers

.903089987 + x (.6989700043) = .84509804 + x (.4771212547)

0.057991947 + x(.6989700043)=x (.4771212547)

is that right so far?

Answer 18

oh lol you were right before, this is no different
fyi the exact numbers go on forever (they are irrational)

anyway go ahead and solve for x by isolating it on one side

Answer 19

haha okay :P

to isolate, what do i do? :/ am I dividing one of them?

Answer 20

oh sorry i missed you already started by combining the constants (correct)

move the .69x to other side by subtracting

Answer 21

0.057991947 + x=x (-0.2218487496)

??

Answer 22

.47x - .69x = -.22x

get rid of the extra "+x" on left side

Answer 23

ohh okay..
what happens next?

Answer 24

how do you solve this

7x = 21

??

Answer 25

divide by 7

x=3 ?

Answer 26

right :)
now apply that to your problem

-.22x = .05

Answer 27

wait so does x=-4.104954486 ?

not sure if that's right? :/

Answer 28

oh wait... is it -.227 ?? so about -0.23 ?

Answer 29

is that right?
x=-0.23 ?

Answer 30

http://www.wolframalpha.com/input/?i=8*5%5Ex+%3D+7*3%5Ex

Answer 31

oh so the solution is actually x=-1.567 ?

Answer 32

x = -.2614

Answer 33

ohh okay

Answer 34

how did you get that? :/ i think i did the math wrong haha

Answer 35

you did it right but you had a rounding error
use the exact numbers right from calculator to avoid rounding error

Answer 36

ohh okay cool! will keep that in mind :)

thanks so much!!! :D