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- https://math.libretexts.org/Courses/Reedley_College/Calculus_I_(Casteel)/01%3A_Functions_and_Graphs/1.05%3A_Exponential_and_Logarithmic_FunctionsThe exponential function \(y=b^x\) is increasing if \(b>1\) and decreasing if \(0<b<1\). Its domain is \((−∞,∞)\) and its range is \((0,∞)\). The logarithmic function \(y=\log_b(x)\) is the inverse of...The exponential function \(y=b^x\) is increasing if \(b>1\) and decreasing if \(0<b<1\). Its domain is \((−∞,∞)\) and its range is \((0,∞)\). The logarithmic function \(y=\log_b(x)\) is the inverse of \(y=b^x\). Its domain is \((0,∞)\) and its range is \((−∞,∞)\). The natural exponential function is \(y=e^x\) and the natural logarithmic function is \(y=\ln x=\log_e x\). Given an exponential function or logarithmic function in base \(a\), we can make a change of base to convert this function to a
- https://math.libretexts.org/Courses/Borough_of_Manhattan_Community_College/MAT301_Calculus_I/01%3A_Review-_Functions_and_Graphs/1.06%3A_Exponential_and_Logarithmic_FunctionsWe use the properties of these functions to solve equations involving exponential or logarithmic terms, and we study the meaning and importance of the number \(e\).We also define hyperbolic and invers...We use the properties of these functions to solve equations involving exponential or logarithmic terms, and we study the meaning and importance of the number \(e\).We also define hyperbolic and inverse hyperbolic functions, which involve combinations of exponential and logarithmic functions. (Note that we present alternative definitions of exponential and logarithmic functions in the chapter Applications of Integrations, and prove that the functions have the same properties with either definiti…
- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_210_Calculus_I_(Professor_Dean)/1%3A_Functions_and_Graphs_(Review)/1.5%3A_Exponential_and_Logarithmic_FunctionsWe use the properties of these functions to solve equations involving exponential or logarithmic terms, and we study the meaning and importance of the number \(e\).We also define hyperbolic and invers...We use the properties of these functions to solve equations involving exponential or logarithmic terms, and we study the meaning and importance of the number \(e\).We also define hyperbolic and inverse hyperbolic functions, which involve combinations of exponential and logarithmic functions. (Note that we present alternative definitions of exponential and logarithmic functions in the chapter Applications of Integrations, and prove that the functions have the same properties with either definiti…
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/04%3A_Transcendental_Functions/4.11%3A_Hyperbolic_FunctionsThe hyperbolic functions appear with some frequency in applications, and are quite similar in many respects to the trigonometric functions. This is a bit surprising given our initial definitions.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_400%3A_Calculus_I_-_Differential_Calculus/01%3A_Critical_Concepts_for_Calculus/1.07%3A_Hyperbolic_FunctionsThe material in this section is likely not review. Instead, it introduces an important family of functions called the hyperbolic functions. These functions are used throughout calculus and differentia...The material in this section is likely not review. Instead, it introduces an important family of functions called the hyperbolic functions. These functions are used throughout calculus and differential equations.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/11%3A_A_-_Calculus_Review_-_What_Do_I_Need_to_Know_From_Calculus%3F/11.03%3A_DerivativesIn your calculus classes you have also seen that some relations are represented in parametric form. However, there is at least one other set of elementary functions, which you should already know abou...In your calculus classes you have also seen that some relations are represented in parametric form. However, there is at least one other set of elementary functions, which you should already know about. These are the hyperbolic functions. Such functions are useful in representing hanging cables, unbounded orbits, and special traveling waves called solutions. They also play a role in special and general relativity.
- https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/Interactive_Calculus_Q1/01%3A_Functions_and_Graphs/1.06%3A_Exponential_and_Logarithmic_FunctionsThe exponential function \(y=b^x\) is increasing if \(b>1\) and decreasing if \(0<b<1\). Its domain is \((−∞,∞)\) and its range is \((0,∞)\). The logarithmic function \(y=\log_b(x)\) is the inverse of...The exponential function \(y=b^x\) is increasing if \(b>1\) and decreasing if \(0<b<1\). Its domain is \((−∞,∞)\) and its range is \((0,∞)\). The logarithmic function \(y=\log_b(x)\) is the inverse of \(y=b^x\). Its domain is \((0,∞)\) and its range is \((−∞,∞)\). The natural exponential function is \(y=e^x\) and the natural logarithmic function is \(y=\ln x=\log_e x\). Given an exponential function or logarithmic function in base \(a\), we can make a change of base to convert this function to a
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/06%3A_Techniques_of_Integration/6.06%3A_Hyperbolic_FunctionsThe hyperbolic functions are a set of functions that have many applications to mathematics, physics, and engineering. Among many other applications, they are used to describe the formation of satellit...The hyperbolic functions are a set of functions that have many applications to mathematics, physics, and engineering. Among many other applications, they are used to describe the formation of satellite rings around planets, to describe the shape of a rope hanging from two points, and have application to the theory of special relativity. This section defines the hyperbolic functions and describes many of their properties, especially their usefulness to calculus.
- https://math.libretexts.org/Courses/Prince_Georges_Community_College/MAT_2410%3A_Calculus_1_(Beck)/01%3A_Functions_and_Graphs/1.06%3A_Exponential_and_Logarithmic_FunctionsThe exponential function \(y=b^x\) is increasing if \(b>1\) and decreasing if \(0<b<1\). Its domain is \((−∞,∞)\) and its range is \((0,∞)\). The logarithmic function \(y=\log_b(x)\) is the inverse of...The exponential function \(y=b^x\) is increasing if \(b>1\) and decreasing if \(0<b<1\). Its domain is \((−∞,∞)\) and its range is \((0,∞)\). The logarithmic function \(y=\log_b(x)\) is the inverse of \(y=b^x\). Its domain is \((0,∞)\) and its range is \((−∞,∞)\). The natural exponential function is \(y=e^x\) and the natural logarithmic function is \(y=\ln x=\log_e x\). Given an exponential function or logarithmic function in base \(a\), we can make a change of base to convert this function to a
- https://math.libretexts.org/Courses/City_University_of_New_York/Calculus_I_(CUNY)/01%3A_Functions_and_Graphs/1.06%3A_Exponential_and_Logarithmic_FunctionsThe exponential function \(y=b^x\) is increasing if \(b>1\) and decreasing if \(0<b<1\). Its domain is \((−∞,∞)\) and its range is \((0,∞)\). The logarithmic function \(y=\log_b(x)\) is the inverse of...The exponential function \(y=b^x\) is increasing if \(b>1\) and decreasing if \(0<b<1\). Its domain is \((−∞,∞)\) and its range is \((0,∞)\). The logarithmic function \(y=\log_b(x)\) is the inverse of \(y=b^x\). Its domain is \((0,∞)\) and its range is \((−∞,∞)\). The natural exponential function is \(y=e^x\) and the natural logarithmic function is \(y=\ln x=\log_e x\). Given an exponential function or logarithmic function in base \(a\), we can make a change of base to convert this function to a
- https://math.libretexts.org/Courses/College_of_Southern_Nevada/Calculus_(Hutchinson)/07%3A_Integrals_and_Transcendental_Functions/7.03%3A_Hyperbolic_FunctionsCertainly the hyperbolic functions do not closely resemble the trigonometric functions graphically. But they do have analogous properties.
