Exponential and Log Functions

Example

Total cost: \(C=C(Q)\)

Marginal cost: \(MC = C'(Q)\)

Average cost: \[AC = \frac{C(Q)}{Q}\]

When is \(\dfrac{d AC}{d Q}\) positive?

Example

Revenue: \(R=f(Q)\)

Output: \(Q = g(L)\)

How does revenue change due to a change in labor input \(L\)? \[\frac{d R}{d L} = \frac{d R}{d Q} \cdot \frac{d Q}{d L}\]

Exponential Functions

The exponential function can be represented as: \[y=f(x)=b^{x} \quad (b>1)\] where \(b\) denotes a fixed base of the exponent.

A more generalized version can be written as: \[y=a b^{c x}\]

Natural Exponential Function

Natural exponential function: the base is a special mathematical constant called Euler’s number \(e=2.71828...\) \[y=a e^{r x}\]

Natural Exponential Function

Jacob Bernoulli discovered this constant in 1683 while studying a question about compound interest.

Invest \(A\) dollars at a 5% interest rate that compounds yearly. After one year you have \(A(1+0.05)\); after \(t\) years: \[A(1+0.05)^{t}\]

Natural Exponential Function

After \(t\) years: \(A(1+0.05)^{t}\). Rewrite this as: \[A\Bigg(\underbrace{\left(1+\frac{1}{1/0.05}\right)^{1/0.05}}_{\approx\, e}\Bigg)^{0.05t} \approx A e^{0.05t}\]

More generally, with initial principal \(A_0\), interest rate \(r\), and horizon \(t\): \[A_t \approx A_0 e^{rt}\]

Logarithmic Function

Since the exponential function is a monotonic function, its inverse exists.

The inverse of the exponential function is called the log or logarithmic function.

For the exponential function: \[y=b^{x} \rightarrow \log_b y = x\]

For the natural exponential function: \[y=e^{x} \rightarrow \log_{e} y = \ln y = x\]

\(y = e^x\)

\(y = \ln x\)

Rules for Logarithmic Functions

Let \(p=\ln u\) and \(q=\ln v\), so that \(u=e^{p}\) and \(v=e^{q}\).

\(\ln (u v)=\ln u+\ln v\), since \[uv=e^{p}e^{q}=e^{p+q} \;\Rightarrow\; \ln(uv)=p+q\]

\(\ln (u / v)=\ln u-\ln v\), since \[\frac{u}{v}=\frac{e^{p}}{e^{q}}=e^{p-q} \;\Rightarrow\; \ln\left(\frac{u}{v}\right)=p-q\]

\(\ln u^{a}=a \ln u\), since \[u^{a}=\left(e^{p}\right)^{a}=e^{ap} \;\Rightarrow\; \ln u^{a}=ap\]

Derivatives of Exponential Functions

Derivative of the exponential function: \[y=e^{x} \quad \rightarrow \quad \frac{dy}{dx} = e^x\]

Using the chain rule: \[y=e^{f(x)} \quad \rightarrow \quad \frac{dy}{dx} = f'(x) e^{f(x)}\]

Derivatives of Logarithmic Functions

Derivative of the log function: \[\frac{d}{d x} \ln x =\frac{1}{x}\]

Using the chain rule: \[\frac{d}{d x} \ln f(x) =\frac{f'(x)}{f(x)}\]

Examples

Find the derivatives for the following functions:

1. \(y=e^x\)
2. \(y=\ln x\)
3. \(y=ae^{rx}\)
4. \(y=e^{-x}\)
5. \(y=\ln ax\)
6. \(y=\ln x^{c}\)

Homework Questions

• Exercise 10.5: 1, 3, 7

Textbook reference: 10.5

Notes on Log and Exponential Functions are on the website