Help with 3.4 Exponential and logarithmic Equations and inequalities

Question

Provide a video on how you were able to get that answer and steps

Solve each Equations

1.) 8^x = 32 ---------------------> answer: x= 5/3

2.) log₂ x = -1 -----------------> answer: x= 1/2

Provide a video on how you were able to get that answer and steps

solve each exponential equation. write the exact answer with natural logarithms and then approximate the result correctly to three decimal places.

1.) 2^x = 3 -----------------> answer: x = In 3 / In 2 = 1.585

2.) 2^2x+3 = 15 ----------------> answer: X= (In 15 - 3 In 2) / 2 In 2 = 0.453

3.) e^x+1 = 3 ---------------> answer:x = -1 + In 3 = 0.099

4.) 2e^x-2+3 = 7 ---------------> answer: x= In 10 - In 3 + In 4 / 2 In 4 = 0.934

5.) 5^1-x = 2^x --------------> answer: x = In 5 / In 5 + In 2 = 0.699

6.) 2^1-x = 3^4x+6 ------------> answer:x= In 2 - 6 In 3 / 4 In 3 + In 2 = -1.159

Answer 1

I'll work through one of the questions by text now. I may try a some others (maybe even a video) later when I have more time.

I picked question 2 (in the second part), which looks like it might require the most work, as one that might illustrate some useful ideas.

2.) 2^2x+3 = 15 ----------------> answer: x = (In 15 – 3 In 2) / 2 In 2 = 0.453

Step 1: Convert the left hand side to a base e exponential

Since 2 = e^ln2 this means that 2^2x+3 = (e^ln2)^2x+3 = e^(2x+3)ln2

Step 2: Substitution: on the left hand side replace 2^2x+3 with e^(2x+3)ln2

e^(2x+3)ln2 = 15

Step 3: Take the natural logarithm (ln) of both sides

ln(e^(2x+3)ln2) = ln15 which means that (2x + 3)ln2 = ln15

since ln(x) is the inverse function of e^x, ln(e^x) = x

Step 4: Divide both sides by ln2

(2x +3)ln2/ln2 = ln15/ln2 which means that 2x + 3 = ln15/ln2

Step 5: Subtract 3 from both sides

2x + 3 – 3 = ln15/ln2 – 3 which means that 2x = ln15/ln2 – 3

Step 6: Convert the right hand side to a single fraction

ln15/ln2 – 3 = ln15/ln2 – 3ln2/ln2 = (ln15 – 3ln2)/ln2

Step 7: Substitution: on the right hand side replace ln15/ln2 – 3 with (ln15 – 3ln2)/ln2

2x = (ln15 – 3ln2)/ln2

Step 8: Divide both sides by 2

2x/2 = [(ln15 – 3ln2)/ln2]/2

which means that

x = (ln15 – 3ln2)/2ln2 = 0.453 (rounded to three decimal places)