Exponential functions and logarithmic functions are very special functions, and so are their derivatives.

Use tables to graph \(f(x)=2^x\) and \(g(x)=\left(\dfrac{1}{2}\right)^x\)

Graph: \(f(t)=e^{0.3t}\); \(f(t)=e^{1t}\); \(f(t)=e^{2t}\)

Graph: \(f(t)=e^{-0.3t}\); \(f(t)=e^{-1t}\); \(f(t)=e^{-2t}\)

Using the exponential function \(A(t)=5e^{0.06931t}\) predict how many bacteria will be on the counter after \(2\) hours.

If in a discussion group of \(10\) people, the first-ranked person participated \(35\) times, use the exponential function \[N(p)=N_1e^{-0.11(p-1)}\quad 1\leq p\leq 10\] to determine how many times the sixth-ranked person participates.

Use the function$$A(t)=P{\left(1+\genfrac{}{}{0ex}{}{r}{n}\right)}^{nt}$$to determine the amount of money accumulated after $15$ years if $\$2\text{,}000$ is invested in an account that pays $8\%$ interest compounded quarterly.

Logarithmic functions are very closely related to exponential functions, both in their definitions and their inverse functions.

Convert each exponential function to the corresponding logarithmic function.

1. \(y=e^{2x}\)

2. \(a=e^{4x+3}\)

3. \(20=e^{-0.3x}\)

Convert each logarithmic function to the corresponding exponential function.

1. \(5=\ln{(7x)}\)

2. \(x+5=\ln{y-3}\)

3. \(-0.01=\ln{x+8}\)

Natural logarithms, two special identities, and a postage stamp.

Evaluate each of the following:

1. \(\ln e\)

2. \(\ln 1\)

Use the properties of the natural logarithm to expand each logarithmic expression.

1. \(\ln(3x)\)

2. \(\ln\left(\dfrac{8x}{x+4}\right)\)

3. \(\ln x^6\)

4. \(12\text{,}000\ln\left(xy^5\right)\)

Use the properties of the natural logarithm to write each logarithmic expression as an expression with a single logarithm.

1. \(\ln x+6\ln y\)

2. \(\ln a-\ln b -\ln c\)

To four decimal places, approximate the solution to the equation \(e^{2x+5}=12\).

Justifying the derivative for the natural logarithm function. An intuitive look, then the actual derivation.

For the function \(f(x)=7x^3\ln{x}\), find

1. \(f'(x)\)

2. an equation of the line tangent at \(x=e\).

Find \(f'(x)\) for \(f(x)=\ln{\left(5x^2+2x-7\right)}\).

Find \(f'(x)\) for \(f(x)=\ln\left(x^2-4\right)^3\)

Find \(f'(x)\) for \(f(x)=[\ln\left(3x\right)]^4\)

Find \(f'(x)\) for \(f(x)=\dfrac{6}{\ln\left(8x^2\right)}\)

Find \(y'\) for \(3x^4y^2+\ln\left(xy^2\right)=6\)

Explaining the derivative of the natural exponential function.

Find \(f'(x)\) for \(f(x)=8x^3e^x\).

Find the equation of the line tangent to the curve \(f(x)=7xe^x\) at the point \((0,0)\). Round decimals to two places.

Find \(f'(x)\) for \(f(x)=e^{3x^2+5x}\).

A little more about natural exponential functions, their derivatives, and equations of tangent lines.

Find \(f'(3)\) for \(f(x)=\ln\left(2x+e^{-0.05x}\right)\). Round the result to \(4\) decimal places.

Find \(f''(x)\) for \(f(x)=e^{2x^2+5}\)

Find \(f'(x)\) for \(f(x)=6e^{-7x}\ln 4x\)

Find \(f'(0)\) for \(f(x)=\dfrac{e^x-e^{-x}}{e^x+e^{-x}}\)

Find the derivative.

1. \(\dfrac{d}{dx}\left[e^{4x}\right]\)

2. \(\dfrac{d}{dx}\left[e^{-6x}\right]\)

3. \(\dfrac{d}{dx}\left[5e^{-0.2x}\right]\)