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Mathematics LibreTexts

4.3: Other Bases

  • Page ID
    532
  • [ "article:topic", "authorname:green" ]

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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

    \[ f(x)=2^x. \]

    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?

    Larry Green (Lake Tahoe Community College)

    • Integrated by Justin Marshall.