- Radicals & rational exponents Examples, solutions, videos, worksheets, and activities to help PreCalculus students learn about exponential and logarithmic functions. Let \(u=2x^3\) and \(du=6x^2\,dx\). If the initial population of fruit flies is \(100\) flies, how many flies are in the population after \(10\) days? Exponential functions and logarithm functions are important in both theory and practice. - Graphing logarithmic functions Example \(\PageIndex{5}\): Evaluating a Definite Integral Involving an Exponential Function, Evaluate the definite integral \(\displaystyle ∫^2_1e^{1−x}\,dx.\), Again, substitution is the method to use. \[\begin{align*} ∫x^{−1}\,dx &=\ln |x|+C \\[4pt] ∫\ln x\,\,dx &= x\ln x−x+C =x (\ln x−1)+C \\[4pt] ∫\log_a x\,dx &=\dfrac{x}{\ln a}(\ln x−1)+C \end{align*}\], Example \(\PageIndex{9}\): Finding an Antiderivative Involving \(\ln x\), Find the antiderivative of the function \(\dfrac{3}{x−10}. Today, logarithms are still important in many fields of science and engineering, even though we use calculators for most simple calculations. 0000002760 00000 n
Example \(\PageIndex{2}\): Square Root of an Exponential Function. Let \(u=x^4+3x^2\), then \(du=(4x^3+6x)\,dx.\) Alter \(du\) by factoring out the \(2\). Thus, \[∫3x^2e^{2x^3}\,dx=\frac{1}{2}∫e^u\,du. Unit: Exponential & logarithmic functions, Multiplying & dividing powers (integer exponents), Powers of products & quotients (integer exponents), Multiply & divide powers (integer exponents), Properties of exponents challenge (integer exponents), Exponential equation with rational answer, Rewriting quotient of powers (rational exponents), Rewriting mixed radical and exponential expressions, Properties of exponents intro (rational exponents), Properties of exponents (rational exponents), Evaluating fractional exponents: negative unit-fraction, Evaluating fractional exponents: fractional base, Evaluating quotient of fractional exponents, Simplifying cube root expressions (two variables), Simplifying higher-index root expressions, Simplifying square-root expressions: no variables, Simplifying rational exponent expressions: mixed exponents and radicals, Simplifying square-root expressions: no variables (advanced), Worked example: rationalizing the denominator, Simplifying radical expressions (addition), Simplifying radical expressions (subtraction), Simplifying radical expressions: two variables, Simplifying radical expressions: three variables, Simplifying hairy expression with fractional exponents, Exponential expressions word problems (numerical), Initial value & common ratio of exponential functions, Exponential expressions word problems (algebraic), Interpreting exponential expression word problem, Interpret exponential expressions word problems, Writing exponential functions from tables, Writing exponential functions from graphs, Analyzing tables of exponential functions, Analyzing graphs of exponential functions, Analyzing graphs of exponential functions: negative initial value, Modeling with basic exponential functions word problem, Exponential functions from tables & graphs, Rewriting exponential expressions as A⋅Bᵗ, Equivalent forms of exponential expressions, Solving exponential equations using exponent properties, Solving exponential equations using exponent properties (advanced), Solve exponential equations using exponent properties, Solve exponential equations using exponent properties (advanced), Interpreting change in exponential models, Constructing exponential models: half life, Constructing exponential models: percent change, Constructing exponential models (old example), Interpreting change in exponential models: with manipulation, Interpreting change in exponential models: changing units, Interpret change in exponential models: with manipulation, Interpret change in exponential models: changing units, Linear vs. exponential growth: from data (example 2), Comparing growth of exponential & quadratic models, Relationship between exponentials & logarithms, Relationship between exponentials & logarithms: graphs, Relationship between exponentials & logarithms: tables, Evaluating natural logarithm with calculator, Using the properties of logarithms: multiple steps, Proof of the logarithm quotient and power rules, Evaluating logarithms: change of base rule, Proof of the logarithm change of base rule, Logarithmic equations: variable in the argument, Logarithmic equations: variable in the base, Solving exponential equations using logarithms: base-10, Solving exponential equations using logarithms, Solving exponential equations using logarithms: base-2, Solve exponential equations using logarithms: base-10 and base-e, Solve exponential equations using logarithms: base-2 and other bases, Exponential model word problem: medication dissolve, Exponential model word problem: bacteria growth, Transforming exponential graphs (example 2), Graphs of exponential functions (old example), Graphical relationship between 2ˣ and log₂(x), This topic covers: 0000001865 00000 n
Integral formulas for other logarithmic functions, such as \(f(x)=\ln x\) and \(f(x)=\log_a x\), are also included in the rule. Example \(\PageIndex{12}\): Evaluating a Definite Integral, Evaluate the definite integral \[∫^{π/2}_0\dfrac{\sin x}{1+\cos x}\,dx.\nonumber\], We need substitution to evaluate this problem. This topic covers: - Radicals & rational exponents - Graphs & end behavior of exponential functions - Manipulating exponential expressions using exponent properties - Exponential growth & decay - Modeling with exponential functions - Solving exponential equations - Logarithm properties - Solving logarithmic equations - Graphing logarithmic functions - Logarithmic scale Then, divide both sides of the \(du\) equation by \(−0.01\). Integrating functions of the form \(f(x)=x^{−1}\) result in the absolute value of the natural log function, as shown in the following rule. Using the equation \(u=1−x\), we have: \[\text{When }x = 1, \quad u=1−(1)=0, \nonumber\], \[\text{and when }x = 2, \quad u=1−(2)=−1. Expressed mathematically, x is the logarithm of n to the base b if b x = n, in which case one writes x = log b n.For example, 2 3 = 8; therefore, 3 is the logarithm of 8 to base 2, or 3 = log 2 8. \(\displaystyle ∫e^x(3e^x−2)^2\,dx=\dfrac{1}{9}(3e^x−2)^3+C\), Example \(\PageIndex{3}\): Using Substitution with an Exponential Function, Use substitution to evaluate the indefinite integral \(\displaystyle ∫3x^2e^{2x^3}\,dx.\). \nonumber\], If the supermarket sells \(100\) tubes of toothpaste per week, the price would be, \[p(100)=1.5e−0.01(100)+1.44=1.5e−1+1.44≈1.99. \nonumber\], The next step is to solve for \(C\). We cannot use the power rule for the exponent on \(e\). Find the antiderivative of the function using substitution: \(x^2e^{−2x^3}\). Download for free at http://cnx.org.