Question

Solve the following exponential equation. Express your answer as both an exact expression and a decimal approximation rounded to two decimal places.

3e^4x = 63

Answer 1

If you wish to enter log or ln, you must use the keypad.

x = _____________ ≈ __________

Answer 2

is it 3e^(4x) = 6^3? or 63? or can you take a screencap?

Answer 3

this one is one of the simpler ones

divide both sides by 3

e^(4x) = 63/3

take the natural log of both sides to eliminate e, divide both sides by 4 to isolate x

Answer 4

x = ln(21)/4

Answer 5

perfect

for the decimal part, we can just chug it into a calculator and get x = 0.76113 which rounds to 0.76

Answer 6

Solve the following logarithmic equation. Express your answer as either an exact expression or a decimal approximation rounded to four decimal places. If there is no solution, indicate "No Solution (∅)."

log_6(x−4)+log_6(x+1)=log_6(x+3)

If you wish to enter log or ln, you must use the keypad. If there is more than one solution, separate your answers with a comma.

Selecting a radio button will replace the entered answer value(s) with the radio button value. If the radio button is not selected, the entered answer is used.

x = ____________________ No Solution (∅)

Answer 7

log rules - when you have two logs of the same base being added together, you can combine them, multiplying what's inside

in other words:

log_6(x−4)+log_6(x+1) ---> combine these two log base 6, multiply what's inside both logs ---> log_6( (x-4)*(x+1) )

setting both sides equal again

log_6( (x-4)*(x+1) ) = log_6(x+3)

notice how you have the same log base 6 on both sides, so you can raise both sides to the base of 6 and eliminate all the logs

(x-4)(x+1) = (x+3)

FOIL the left side, combine terms, either factor or use the quadratic formula to get your x-values

Answer 8

\[x = 2 + {\sqrt{11}}\]

Answer 9

almost, for this quadratic equation we have 2 solutions (when you use the quadratic formula there's a +/- sign)

x = 2 + sqrt(11)
and
x = 2 - sqrt(11)

Answer 10

that's true

Answer 11

it said wrong

The answer you submitted, 2+√11,2−√11, is incorrect.

It appears you have correctly used the quadratic formula to calculate possible solutions, but have made an error substituting these values back into log_6(x−4)+log_6(x+1)=log_6(x+3) to check them.

You may wish to check your work.

For a step-by-step guide to solving this problem, select Step By Step.

Answer 12

i can still do it

Answer 13

ahh sorry, I forgot - logs can only take positive values in their input

2 - sqrt(11) doesn't work because it would make log(x+1) negative

so it's only 2 + sqrt(11)

Answer 14

it's okay

Answer 15

correct

Answer 16

another one is the same but different

Answer 17

Solve the following logarithmic equation. Express your answer as either an exact expression or a decimal approximation rounded to four decimal places. If there is no solution, indicate "No Solution (∅)."

log_4(x+1)−log_4(x−1)=2

Answer 18

x = 17/15

Answer 19

is that right

Answer 20

yup that's what I got too

Answer 21

i knew it

Answer 22

Solve the following exponential equation. Express your answer as both an exact expression and a decimal approximation rounded to two decimal places.

2^e2x + 10 = 97

x = _____________ ≈ __________

ln(97/2) / 2 -5

≈ __________

-3.05921810

Answer 23

can you take a screengrab of the equation?

Answer 24

ummm i don't think so , my phone acting stupid but i could try

Answer 25

can you try adding parentheses around the exponents? I just need to know what's in the exponent or not

Answer 26

right , of course. excuse me one moment

Answer 27

i'm sorry if it's blurry , i had to use an ipad

Answer 28

i tried to focus it

Answer 29

good just remember to round to 2 places

-3.059 rounds up to -3.06

Answer 30

Solve the following logarithmic equation. Express your answer as either an exact expression or a decimal approximation rounded to four decimal places. If there is no solution, indicate "No Solution (∅)."

ln(x−4)=ln(5x)

Answer 31

they're both ln, so you can raise both sides to base e and eliminate the logs

(x-4) = 5x, should be straightforward from there

Answer 32

i think it's no solution

Answer 33

good, since x is negative ln(5x) becomes undefined, so that's correct

Answer 34

Solve the following logarithmic equation. Express your answer as either an exact expression or a decimal approximation rounded to four decimal places. If there is no solution, indicate "No Solution (∅)."

log_3(2x−5)+log_3(x)=log_3(5)

Answer 35

\[x = \frac{ 5 + \sqrt{65}}{ 4}\]

Answer 36

good

Answer 37

Solve the following exponential equation. Express your answer as both an exact expression and a decimal approximation rounded to two decimal places. Use e=2.71828182845905.

e^2x+3 = 4^3x/7

x = _____________ ≈ __________

Answer 38

-x = -21/14-ln(64)

-2.13390412

are these right

Answer 39

yeah just make sure to round to -2.13

Answer 40

Solve the following exponential equation. Express your answer as both an exact expression and a decimal approximation rounded to two decimal places. Use e=2.71828182845905.

e^2x−2 = 132^4x+8

x = _____________ ≈ __________

Answer 41

-x = -1+4 ln(132)/2ln(132) - 1

-2.34224681

Answer 42

I think you made a sign error in the left part

should be -1 - 4ln(132) in the numerator

other than that, good, make sure to round the decimal part -2.34

Answer 43

okay you might be right

Answer 44

no , i was right with the +

Answer 45

Damon is saving up money for a down payment on a motorcycle. He currently has $5371, but knows he can get a loan at a lower interest rate if he can put down $6197. If he invests the $5371 in an account that earns 4.2% annually, compounded monthly, how long will it take Damon to accumulate the $6197? Round your answer to two decimal places, if necessary.

Answer 46

A = P(1 + r/n)^(nt)

P is the principal (5371), r is the interest rate as a decimal (4.2%, so r = 0.042), n is the # of times compounded per year (monthly compounding, so n = 12), t is time (which you'll solve for). plug in and solve for t.

Answer 47

oh right, since he wants $6197, set A = 6197

Answer 48

6197 years ?

Answer 49

no, t is time which is what you want to solve for. A is the amount he wants.

set A = 6197 in the equation and solve for t.

Answer 50

Oh sorry

Answer 51

how could i set it up

Answer 52

A = 6197 and 12 is the time right

Answer 53

the problem is asking how long the account will take to reach 6197, which means time is unknown, meaning you need to solve for t

A = P(1 + r/n)^(nt)

A = the final amount = 6197
P = principal = 5371
r = interest rate = 0.042
n = # of times compounded per year = 12

plug these in and solve for t

Answer 54

5371(1 + 0.042/12)^(12t) ?

Answer 55

is that right ?

Answer 56

yeah but you'd set that equal to the A-value, 6197

6197 = 5371(1 + 0.042/12)^(12t)

Answer 57

oh okay

Answer 58

t = 3.41193778

Answer 59

good just make sure to round to 3.41

Answer 60

Tyron is saving up money for a down payment on a motorcycle. He currently has $3256, but knows he can get a loan at a lower interest rate if he can put down $3988. If he invests the $3256 in an account that earns 5.6% annually, compounded continuously, how long will it take Tyron to accumulate the $3988? Round your answer to two decimal places, if necessary.

Answer 61

continuous compounding is a bit different, there's a different formula

A = Pe^(rt)

A is the final amount (3988), P is the principal (3256), r is the interest rate (0.056), e is euler's constant, t is time, solve for t

Answer 62

A = P(1 + r/n)^(nt)

A = the final amount = 3988
P = principal = 3256
r = interest rate = 0.056

Answer 63

3988 = 3256(1 + 0.056/12)^(12t)

Answer 64

this is a continuous compounding problem, so the formula is different

Answer 65

3988 = 3256e^(0.056t)

is that right

Answer 66

yes

Answer 67

t = 3.62125721

Answer 68

is that right

Answer 69

yeah just be sure to round to 3.62

Answer 70

In chemistry, the pH of a solution is a measure of the acidity or alkalinity of a solution. Water has a pH of 7 and, in general, acids have a pH less than 7 and alkaline solutions have a pH greater than 7. Find the pH of a solution with a hydronium ion concentration of 5.1×10^−6 moles/liter. Round your answer to two decimal places, if necessary.

pH =

Answer 71

pH is -log(hydronium concentration), so it's just -log(5.1×10^−6), plug into a calculator and round appropriately