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Mathematics LibreTexts

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.