Question

use the natural log to find x... 5^x=2e^x+1... answer in the back of the book is (1+ln2)/(-1+ln5).. but not sure how to get there.. please show work!!!!!!!!

Answer 1

@phi

Answer 2

is the x+1 the exponent ?

Answer 3

yes

Answer 4

i had the idea of taking the natural log of both sides

Answer 5

that sounds promising. what do you get ?

Answer 6

ln5^x=2(x+1)

Answer 7

right?

Answer 8

you should do it in steps
\[ 5^x=2e^{x+1} \\
\ln\left( 5^x\right) = \ln\left( 2e^{x+1}\right)
\]

Answer 9

cant i just cross out the ln and e on the right though

Answer 10

almost. but you have a 2 in there. use log(a*b) = log(a) + log(b) on the right side
on the left side use
log(a^b) = b*log(a)

Answer 11

so on left side it would be.. xlog(5)

Answer 12

yes. but write out the whole equation.

Answer 13

im not sure about the right side..

Answer 14

use log(a*b) = log(a) + log(b)
where a matches 2 and b matches the e^stuff

Answer 15

okay so...... xlog5=log2+log(e^x+1)

Answer 16

and use parens , because (x+1) is the exponent:
xlog5=log2+log(e^(x+1) )

now you can "cross out the log and the e"

Answer 17

ok so.... xlog5=log2+(x+1)

Answer 18

yes. solve for x. remember log5 (or maybe we should use ln 5 ? ) is just a constant

Answer 19

can i subtract the x to the left side of the equation

Answer 20

yes

Answer 21

but wouldnt that cancelout the x's

Answer 22

write down what you get

Answer 23

log5=log2+1

Answer 24

xlog5=log2+(x+1)
or
x log5 = log2 + x + 1
add -x to both sides

Answer 25

ya and i got log5=log2+1

Answer 26

you get
x log5 -x = log2 + x + 1 -x

on the right side x - x is 0 so
x log5 -x = log2 + 1

if this were
1.608*x - x = log2 +1
would the x's disappear ?

notice that ln5 is about 1.608...

Answer 27

what about this idea? factor an x out of each term on the left side?
x ln5 - x = ln2 + 1

Answer 28

ok then what is is the next step

Answer 29

divide by ln5?

Answer 30

factor an x out of each term on the left side?
x ln5 - x = ln2 + 1

what do you get ?

Answer 31

x(ln5-1)=ln2+1

Answer 32

now divide both sides by (ln5 -1)

Answer 33

sweet thanks so much!