4.3: Other Bases
 Page ID
 532
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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

\[x^{2x+1} \]

\[x^4.\]
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
\[S=8014\ln (t+1) \]
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.