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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/Courses/Coastline_College/Math_C170%3A_Precalculus_(Tran)/04%3A_Exponential_and_Logarithmic_Functions/4.04%3A_Logarithmic_FunctionsThe inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.
- https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_1e_(OpenStax)/04%3A_Exponential_and_Logarithmic_Functions/4.03%3A_Logarithmic_FunctionsThe inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Professor_Holz'_Topics_in_Contemporary_Mathematics/04%3A_Population_Growth_Models/4.04%3A_Logarithmic_FunctionsThe inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.
- 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/Supplemental_Modules_(Calculus)/Integral_Calculus/4%3A_Transcendental_Functions/4.1%3A_Logs_and_Derivatives\[ \int_1^x \dfrac{1}{t} dt = \ln x. \nonumber \] \[ \dfrac{d}{dx} \ln x = \dfrac{1}{x}. \nonumber \] \[\dfrac{d}{dx} \ln x^n = \dfrac{d}{dx} \int_1^{x^n} \dfrac{1}{t} dt \nonumber \] \[ = \dfrac{1}{x...\[ \int_1^x \dfrac{1}{t} dt = \ln x. \nonumber \] \[ \dfrac{d}{dx} \ln x = \dfrac{1}{x}. \nonumber \] \[\dfrac{d}{dx} \ln x^n = \dfrac{d}{dx} \int_1^{x^n} \dfrac{1}{t} dt \nonumber \] \[ = \dfrac{1}{x^n} \left( n\,x^{n-1}\right) = n \left(\dfrac{1}{x} \right) \nonumber \] \[ = n \dfrac{d}{dx} \int _1^x \dfrac{1}{t} dt = n\ln x. \nonumber \] \[ \ln x^n = n \ln x + C. \nonumber \] Find the relative extrema of \( x \ln x\).
- https://math.libretexts.org/Courses/Coastline_College/Math_C097%3A_Support_for_Precalculus_Corequisite%3A_MATH_C170/1.05%3A_Exponential_and_Logarithmic_Functions/1.5.04%3A_Logarithmic_FunctionsWe read a logarithmic expression as, “The logarithm with base \(b\) of \(x\) is equal to \(y\),” or, simplified, “log base \(b\) of \(x\) is \(y\).” We can also say, “\(b\) raised to the power of \(y\...We read a logarithmic expression as, “The logarithm with base \(b\) of \(x\) is equal to \(y\),” or, simplified, “log base \(b\) of \(x\) is \(y\).” We can also say, “\(b\) raised to the power of \(y\) is \(x\),” because logs are exponents. Also, since the logarithmic and exponential functions switch the \(x\) and \(y\) values, the domain and range of the exponential function are interchanged for the logarithmic function.
- 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
