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# 4.3: Exponentials With Other Bases

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## Exponentials With Other Bases

Definition

Let $$a > 0$$ then

$a^x = e^{x \,ln \,a}.$

Example 1

Find the derivative of

$f(x)=2^x.\nonumber$

Solution

We write

$2^x = e^{x \ln 2}.\nonumber$

Now use the chain rule

$f'(x)=(e^{x \ln 2})(\ln 2).\nonumber$

Example 2

Find the derivative of

$f(x)=3^{\sin x}. \nonumber$

Solution

We write

$3^{\sin x}=e^{(\sin x)(\ln 3)}.\nonumber$

Now use the chain rule

$f'(x) =e^{(\sin x)(\ln 3)} (\cos x)(\ln 3).\nonumber$

Example 3

Find the derivative of

$f(x)=x^x. \nonumber$

Solution

We write

$x^x=e^{x\ln x}. \nonumber$

Notice that the product rule gives

$(x \ln x)' = 1 + \ln x. \nonumber$

So using the chain rule we get

\begin{align*} f'(x) &=e^{x \ln x}(1+ \ln x) \\ &= x^x (1+\ln x). \end{align*}

Exercises

Find the derivatives of

1. $$x^{2x+1}$$
2. $$x^4.$$

## Logs With Other Bases

Definition

$\log_a x = \dfrac{\ln\, x}{\ln\, a}. \nonumber$

Example 4

Find the derivative of

$f(x)= \log_4 (x).\nonumber$

Solution

We use the formula

$f(x)=\dfrac{\ln x }{\ln 4} \nonumber$

so that

$f'(x)=\dfrac{1}{x \ln 4}. \nonumber$

Example 5

Find the derivative of

$f(x)= \log (3x+4).\nonumber$

Solution

We again use the formula

$f(u)=\dfrac{\ln(3x+4)}{\ln 10} \nonumber$

now use the chain rule to get

$f'(x)=\dfrac{3\ln (3x+4)}{\ln 10}. \nonumber$

Example 6

Find the derivative of

$f(x)= x\log (2x). \nonumber$

Solution

Use the product rule to get

$f'(x) = \log (2x) +x(\log 2x).\nonumber$

Now use the formula to get

$f'(2x)=\dfrac{\ln 2x}{\ln 10}+\dfrac{x(\ln 2x)'}{\ln 10}.\nonumber$

The chain rule gives

$f'(x)=\dfrac{\ln 2x}{\ln 10}+\dfrac{2x}{2x \ln 2x}\nonumber$

$f'(x)=\dfrac{\ln 2x}{\ln 10} + \dfrac{1}{\ln 10}.\nonumber$

## Integration

Example 7

Find the integral of the following function

$f(x) = 2^x. \nonumber$

Solution

\begin{align*} \int 2^x \; dx &= \int e^{x\ln 2} \; dx \\ u=x\ln 2, \;\; du=\ln 2 \; dx \\ &= \dfrac{1}{\ln 2}\int e^u \; du \\ &= \dfrac{1}{\ln 2} e^u+C \\ &= \dfrac{2^x}{\ln 2} +C. \end{align*}

## Application: Compound Interest

Recall that the interest formula is given by:

$A = P(1 +r/n)^n$

where

• $$n$$ is the number of total compounds before we take the money out,
• $$r$$ is the interest rate,
• $$P$$ is the Principal, and
• $$A$$ is the amount the account is worth at the end.

If we consider continuous compounding, we take the limit as $$n$$ approaches infinity we arrive at

$A=Pe^{rt}.$

Exercise

Students are given an exam and retake the exam later. The average score on the exam is

$S=80-14\ln (t+1) \nonumber$

where $$t$$ is the number of months after the exam that the student retook the exam. At what rate is the average student forgetting the information after 6 months?