Resources for Calculus with other Bases
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Calculus with other Bases Theory
![Calculus with other bases in logarithmic and exponential functions require differentiation and integration concepts.\\ \begin{multicols}{2} \textbf{Differentiation:} $\begin{aligned} & y=\log _a f(x) \\ & y=\frac{\ln f(x)}{\ln a} \rightarrow \frac{d y}{d x}=\frac{1}{\ln a} \times \frac{f^{\prime}(x)}{f(x)} \end{aligned}$\\ \columnbreak \textbf{Integration:} $\begin{aligned} y & =a^x \\ x & =\log_a y \\ & =\frac{\ln y}{\ln a} \text { (change of base) } \\ \ln y & =x \ln a \\ y & =e^{x \ln a}=a^x \\ y^{\prime} & =\ln a e^{x \ln a} \\ & =\ln a a^x \\ &\therefore \displaystyle \int \ln a a^x d x =a^x \\ & \displaystyle \int a^x d x=\frac{1}{\ln a} a^x \end{aligned}$\\ \end{multicols} \begin{multicols}{2} \textbf{Example 1}\\ %10652 Find the equation of the normal to the curve \(y = \log_5(x + 1)\) at \(x= 0\) \\ \textbf{Example 1 solution}\\ $\begin{aligned} y &=\log _{5}(x+1) \\ &=\frac{\log _{e}(x+1)}{\log _{e} 5} \\ y^{\prime} &=\frac{1}{\log _{e} 5(x+1)}\\ \text{At } x=0 \quad y^{\prime}&=\frac{1}{\log_e {5}}\\ M_{N}&=-\log _{e} 5\\ y-0&=-\log_e 5 (x-0)\\ y&=(-\log _{e} 5) x \end{aligned}$\\ \columnbreak \textbf{Example 2}\\ \(\displaystyle \int_1^2 5^x \,d x=?\)\\ \textbf{Example 2 solution}\\ $\begin{aligned} \displaystyle\int_1^2 5^x d x & =\frac{1}{\ln 5}\left[5^x\right]_1^2 \\ & =\frac{1}{\ln 5}\left[5^2-5\right] \\ & =\frac{20}{\ln 5} \end{aligned}$\\ \end{multicols}](/media/ra5dvr5n/4788.png)
NSW Y12 Maths - Advanced Exponential and Log Function Calculus with other Bases
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Calculus with other Bases - Video - Derivative of log with arbitrary base
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