Resources for Calculus with other Bases
-
Questions
11
With Worked SolutionClick Here -
Video Tutorials
1
Click Here
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
Videos
Videos relating to Calculus with other Bases.
-
Calculus with other Bases - Video - Derivative of log with arbitrary base
Plans & Pricing
With all subscriptions, you will receive the below benefits and unlock all answers and fully worked solutions.
-
Teachers TutorsFeaturesFreeProAll ContentAll courses, all topicsQuestionsAnswersWorked SolutionsSystemYour own personal portalQuizbuilderClass ResultsStudent ResultsExam RevisionRevision by TopicPractise ExamsAnswersWorked SolutionsAwesome StudentsFeaturesFreeProContentAny course, any topicQuestionsAnswersWorked SolutionsSystemYour own personal portalBasic ResultsAnalyticsStudy RecommendationsExam RevisionRevision by TopicPractise ExamsAnswersWorked Solutions