Help with these calculus equations!

Question

1) e^4x-3* e^-2x=5e) - I got 2.805

2) log base 8 (x+3) +log base 8 (x+2)=log base 8 (2) - I got 0 as the answer

3) e^2x-15e^x+54 =0 - Was confused (Type an exact answer in simplified form)

4) 5^x+3=3^2x-3- I got 13.822 as the answer (Round to the nearest thousandth)

5) log(13-x)=0.5- I got 9.83772233 as the answer (Type an exact answer, using radicals as needed)

6)lne^x-2lne^4- I got -6 as the answer

7) T=T0+(T1-T0)3^-kt, for t - I was confused

Answer 1

Radikha, it helps all of us to have proper parentheses.,please check the Wyzant Resource “Math: You Need Parentheses”

#1: 1) e^4x-3* e^-2x=5e. I suspect this is written wrong, could you please verify that nothing is missing or excess? For example, after the equal sign you have

5e)

is it really supposed to simply be

5

#2: log base 8 (x+3) +log base 8 (x+2)=log base 8 (2)

I hope you know that log(x) + log(y) = log(x*y), so we can simplify to

log₈((x+3)*(x+2)) = log₈(2)

Then you can raise 8 to the power of each side because log and exponentiation are inverses.

8^{log8((x+3)(x+2))} = 8^log8(2)

(x+3)(x+2) = 2

you can solve this for x.

#3: e^2x-15e^x+54 =0

We need to know which parts are the exponents. I’ll guess

e^2x - 15e^x + 54 = 0

You know how to factor this:

(e^x - 9)(e^x -6) = 0

So the roots are e^x = 9 and e^x = 6

You can take natural log of both sides

Ln(e^x) = ln (9)

x = ln(9)

Similarly x = ln (6)

#4 5^x+3=3^2x-3, I am struggling with the fact that one term is 5^x and the other 3^2x. I would expect them both to have the same base.

#5: log(13-x)=0.5

raise 10 to both sides

10^log(13-x) = 10^.5

13-x = sqrt(10)

x = 13 - sqrt(10)

#6: lne^x-2lne^4

ln(e^x) - 2ln(e⁴)

Remember that log_a a^x = x, and ln means log_e, so

x - 2*4 = x - 8

Is it possible that you didn’t copy #6 exactly right? It isn’t an equation.

#7: T=T0+(T1-T0)3^-kt

This looks like a cooling formula, but it wouldn’t be 3^-kt , but rather e^-kt. Did you copy this problem exactly right?

Answer 2

I didn't match any of your answers and not really sure how to write any of the equations.

Answer 3

1) e^4x-3* e^-2x=5e) - I got 2.805

same base means add exponents

e^(4x-3)+(-2x) = 5e

e^2x-3 = 5e Divide by e means subtract 1 from the exponent

e^(2x-3)-1 = 5

e^2x-4 = 5

ln e^2x-4 = ln 5

2x-4 = ln 5

2x = 4+ln 5

x = 2+(1/2)ln 5 ≈ 2.80

2) log base 8 (x+3) +log base 8 (x+2)=log base 8 (2) - I got 0 as the answer

log₈ (x+3) + log₈ (x+2) = log₈ 2 same base, log + log means multiply insides

log₈ [(x+3)(x+2)] = log₈ 2

x²+5x+6 = 2

x²+5x+4 = 0

x = -4, -1 Since the guts of log must be ≥ 0, only -1 is valid.

3) e^2x-15e^x+54 =0 - Was confused (Type an exact answer in simplified form)

Factor.

(e^x - 9)(e^x - 6) = 0

x = ln 6 or ln 9

4) 5^x+3=3^2x-3- I got 13.822 as the answer (Round to the nearest thousandth)

(x+3)ln 5 = (2x-3)ln 3

xln 5 + 3ln 5 = 2xln 3 - 3ln 3

3ln 5 + 3ln 3 = 2xln 3 - xln 5

(3ln 5 + 3ln 3) / (2ln 3 - ln 5) = x

x ≈ 13.82

5) log(13-x)=0.5- I got 9.83772233 as the answer (Type an exact answer, using radicals as needed)

10^0.5 = 13-x

-x = √10 - 13

x = 13-√10 ≈ 9.8377

6)lne^x-2lne^4- I got -6 as the answer

I might be missing part of the question because I see x-8.

7) T=T0+(T1-T0)3^-kt, for t - I was confused

I think you're supposed to solve it for t.

T = T₀+(T₁-T₀)3^-kt

T-T₀ = (T₁-T₀)3^-kt

(T-T₀) / (T₁-T₀) = 3^-kt

log₃ [(T-T₀) / (T₁-T₀)] = log₃ (3^-kt)

-kt = log₃ [(T-T₀) / (T₁-T₀)]

t = -[log₃ (T-T₀) - log₃ (T₁-T₀)] / k