Exploring Logarithmic Functions

Differentiate: f(x) = ln(x)

• 1/x

• ln(x^2)

• 1/x^2

• e^x

Differentiate: f(x) = log_10(x)

• f'(x) = x / ln(10)

• f'(x) = 1 / x

• f'(x) = ln(10) / x

• f'(x) = 1 / (x * ln(10))

Differentiate: f(x) = ln(3x)

• 3/x

• 1/(3x)

• 1/x

• ln(3)

Differentiate: f(x) = log_2(x^2)

• f'(x) = 2x / (ln(2))

• f'(x) = 1 / (x ln(2))

• f'(x) = 4 / (x ln(2))

• f'(x) = 2 / (x ln(2))

Integrate: ∫ ln(x) dx

• ln(x^2)/2 + C

• x ln(x) + x + C

• x ln(x) - x + C

• x^2/2 + C

Integrate: ∫ log_10(x) dx

• (1/ln(10))(x^2/2 - x) + C

• (1/ln(10))(x ln(x) - x) + C

• (1/ln(10))(x ln(x) + x) + C

• (1/ln(10))(x ln(x) - 2x) + C

Integrate: ∫ x ln(x) dx

• (1/2)x^2 ln(x) - (1/4)x^2 + C

• (1/2)x^2 ln(x) + (1/4)x^2 + C

• (1/3)x^3 ln(x) - (1/9)x^3 + C

• x^2 ln(x) - (1/2)x^2 + C

Integrate: ∫ ln(2x) dx

• x ln(2) - x + C

• x ln(2x) - x + C

• ln(2x^2) + C

• x^2 ln(2x) + C

Differentiate: f(x) = ln(x^3 + 1)

• f'(x) = 3x / (x^3 + 1)

• f'(x) = 3x^2 / (x^3 + 1)

• f'(x) = 3x^2 / (x^2 + 1)

• f'(x) = 1 / (x^3 + 1)

Integrate: ∫ (ln(x))^2 dx

• x(ln(x))^2 - x + C

• x(ln(x))^2 - 3xln(x) + x + C

• x(ln(x))^2 + 2xln(x) + C

• x(ln(x))^2 - 2xln(x) + 2x + C