Chapter 4: Logs and Exponents Review

4.1 Exponential Functions

100

5^x+1=125 (Solve for x)

2

200

5³/5⁶ (Simplify) 

1/5³ or 1/125

300

x^3/4=8 (solve for x)

16

400

(x+3)^4/3=16

5

4.2 Natural Exponential Functions

100

(e³)⁴ (Simplify) 

e¹²

200

e⁵/e^-2 (Simplify)

e⁷

300

e^-1/x =square root of e (Solve for x)

-2

400

3x³=9e³ (Solve for x)

Cube Root of 3e 

4.3 Derivatives of Exponents

100

y = e^1-x (Find Derivative) 

-e¹-x

200

g(x) = e^{square root x} (Find Derivative)

e^{sq root x}/2(sq root x)

300

Determine an equation on the tangent line to the function at a given point:

e^{x^3}, point: (-1, 1/e)

y = (3/e)x + 1/e

400

f(x) = (1+2x)e^4x (Second Derivative) 

2e^4x(3+4x)

4.4 Logarithmic Functions

100

(2/e³)^-1 (Find Derivative)

e³/2

200

e^ln(5x+2)

5x+2

300

ln (3x(x+1)/(2x+1)²) (Expand)

ln3 + lnx + ln (x+1) - 2ln(2x+1)

400

1000 (1+ 0.07/12)^12t =3000 (Solve for t)

t = ln3 / 12ln [1 + 0.07/12] = 15.740

4.5 Derivatives of Logs

100

f(x) = x²(x+1)-3x³ (Find Second Derivative)

-12x+2

200

y = 1/2 (lnx)⁶ (Find Derivative)

3/x(lnx)^5

300

log₄x (Write the Expression with base e)

(ln x)/ (ln 4)