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# 4.3: Other Bases

### Exponentials With Other Bases

Definition

Let $$a > 0$$ then

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

Example 1

Find the derivative of

$f(x)=2^x.$

Solution

We write

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

Now use the chain rule

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

Example 2

Find the derivative of

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

Solution

We write

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

Now use the chain rule

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

Example 3

Find the derivative of

$f(x)=x^x.$

Solution

We write

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

Notice that the product rule gives

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

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

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

Example 4

Find the derivative of

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

Solution

We use the formula

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

so that

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

Example 5

Find the derivative of

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

Solution

We again use the formula

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

now use the chain rule to get

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

Example 6

Find the derivative of

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

Solution

Use the product rule to get

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

Now use the formula to get

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

The chain rule gives

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

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

### Integration

Example 7

Find the integral of the following function

$f(x) = 2^x.$

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

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

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?

### Contributors

• Integrated by Justin Marshall.