4.3: Exponentials With Other Bases
- Page ID
- 532
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
- \(x^{2x+1} \)
- \(x^4.\)
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?
Contributors and Attributions
- Larry Green (Lake Tahoe Community College)
Integrated by Justin Marshall.