Assessment: Logarithmic and Exponential Functions

Evaluate y=4ˣ at x=2.

Evaluate log₅(25).

Find the new horizontal asymptote of y=3ˣ+6 (compared to y=3ˣ, which has asymptote y=0).

Find f'(0) for f(x)=4eˣ.

Find f'(0) for f(x)=3ˣ (using f'(x)=3ˣ·ln3, ln3≈1.0986). Round to 4 decimal places.

Find f'(10) for f(x)=ln(x).

A chemist models radioactive decay using an exponential function and needs the instantaneous rate of decay at a specific time. Which mathematical tool do they need?

A biologist models bacterial population growth as P(t)=P₀×2ᵗ. What does this exponential model assume about the population's growth pattern?

If log_b(x)=3 and b=4, find x.

Find f'(1) for f(x)=eˣ+ln(x). (e≈2.71828) Round to 2 decimal places.