Question

solve equation find exact solution if possible
a)54^-x=1/25
b)e^(3x/4)=10
c)3^(2x)-3x-6=0

Answer 1

for a I got 2.3 and for b I got 3/4 are they right? and I need help with c...

Answer 2

so is the 1st question

\[54^{-x} = \frac{1}{25}\]

Answer 3

ya

Answer 4

well I know for sure and certain is not 2.3

Answer 5

so do you know about logs...?

Answer 6

ya

Answer 7

ok... so take the log of both sides

and you get

\[\ln(54^{-x}) = \log(\frac{1}{25})\]

applying the log laws for powers you get

\[-x \ln(54) = \ln(\frac{1}{25})\]

now solve for x... what do you get..?

Answer 8

so ln of each side ?

Answer 9

yep... that the 1st step

then apply the log law for powers

\[\log_{a}(b^x) = x \log_{a}(b)\]

Answer 10

3.2 ?

Answer 11

nope...

next step... divide both sides by ln(54)

\[-x = \frac{\ln(\frac{1}{25})}{\ln(54)}\]

Answer 12

so I make itt to multipl?

Answer 13

nope just do the division shown...

Answer 14

last step is to multiply both sides of the equation by -1 to find the value of x

Answer 15

so doing div I get -.806

Answer 16

so did you multiply by -1 to get the value of x..?

Answer 17

I had

-x = -0.806

Answer 18

so I feel

x = 0.806

to check find the value of

\[54^{-0.806} \]

does it equal 1/25 or 0.04...?

Answer 19

so the final answer is + 0.806

Answer 20

yep... that's my best guess...

Answer 21

want to take about b..?

Answer 22

yes plz b and c

Answer 23

for b I did Ln10=ln^3x/4 3x/4 = 4/3

Answer 24

ok... when you have the same base... and take the log, you just get the power

\[\log_{a}(a^x) = x\]

so take the base e log or ln of both sides and you get

\[\frac{3x}{4} = \ln(10)\]

so to find x, multiply both sides of the equation by 4

then divide both sides by 3

Answer 25

so if I multiply both by 4 its 3x=ln40 ?

Answer 26

2.5 ?

Answer 27

ok so i got 1.22 @campbell_st

Answer 28

nope if you multiply by 4 you get

\[3x = 4 \times \ln(10)\]

then divide by 3

Answer 29

so x=4/3

Answer 30

yep

\[x = 4 \times \ln(10) \div 3\]

Answer 31

ok :D so how about c do I add the 6 to both sides first?

Answer 32

they want an exact solution for all questions so

a) \[x =- \ln(\frac{1}{25}) \div \ln(54) = -\frac{\ln(1) - \ln(25)}{\ln(54)}\]

and

b) \[x = \frac{4\ln(10)}{3}\]

Answer 33

ok/

Answer 34

is the last equation

\[3^{2x} -3^{x} - 6 = 0\]

Answer 35

yup

Answer 36

ok... this is basically a quadratic equation that can be factored

if I let

\[u = 3^x\]

the equation becomes

\[u^2 - u - 6 = 0\]

which can be factored to

\[(u - 3)(u + 2) = 0\]

so then

the solutions are

u = 3 and u = -2

doing the reverse substitution

\[3^x = -2\]

has no solution...

so the only equation you need to solve is

\[3^x = 3^1\]

what do you think x is...?

Answer 37

1?

Answer 38

thats it... all done

Answer 39

:D thank you

Answer 40

glad to help