November 13th, 2020

Since we’re dealing with a half-life problem, we know the decay factor is .5, since it halves every 6 hours. from a \(\displaystyle \begin{align}100&=50{{e}^{{.085t}}}\\2&=1{{e}^{{.085t}}}\\\ln \left( 2 \right)&={{\cancel{{\ln e}}}^{{.085t}}}\\\ln \left( 2 \right)&=.085t\\t&=\frac{{\ln \left( 2 \right)}}{{.085}}\approx 8.15\end{align}\). Exponential Function Logarithmic Function, \(x={{b}^{y}}\) \(\Leftrightarrow \) \(y={{\log }_{b}}x\). %���� The population is expected to grow by the function \(p\left( … For log and ln functions, use –1, 0, and 1 for the \(y\) values for the parent function. Remember that half-life problems deal with exponential decays that halve for every time period. It’s always a good idea to make sure your answer makes sense (the amount of 40 grams should divide by 2 about \(\displaystyle \frac{{26}}{6}\) or around 4 times to get down to 2). Note: If there is no \(b\) next to the log, then the base is assumed to be 10. is not a high enough rating to have been a moderate quake; the event was probably Here’s a practical example where we can use logs to solve the algebraic equation \(250=100{{\left( {1.05} \right)}^{t}}\) to get the number of years, \(t\), that it would take to go from $100 to $250 with a yearly interest rate of 5%. endobj "Logarithmic Word Problems." b) You test (We can do this since if a number equals another number, their respective logs are the same). Since the half-life (decay interval) is 6, but we want to know how much of the substance there will be after 18 hours, our “time” is actually\(\displaystyle \frac{{18}}{6}\,\) or \(\displaystyle \frac{{\text{time we are interested in}}}{{\text{time of a half-life}}}\), which is 3. Again, the base “\(e\)” has many applications in both engineering and economics. Logarithmic word problems, in my experience, generally involve evaluating a given logarithmic equation at a given point, and solving for a given variable; they're pretty straightforward. so: pH = log[H+] Then change the \(+\) on the outside to a “times” on the inside. eval(ez_write_tag([[970,250],'shelovesmath_com-leader-2','ezslot_11',135,'0','0']));We can solve half-life problems using two different methods; we’ll use both methods here. At the end of 2017, there were 15,000 residents. Suppose that a graduating class had 500 students graduating the first year, but after that, the number of students graduating declines by a certain percentage. <>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 612 792] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> These are powerful properties that we’ll need to use to isolate variables in exponents so we can solve for them. Then, because of the \(\frac{1}{3}\), take the cube root of the argument. Multiplying and Dividing, including GCF and LCM, Powers, Exponents, Radicals (Roots), and Scientific Notation, Introduction to Statistics and Probability, Types of Numbers and Algebraic Properties, Coordinate System and Graphing Lines including Inequalities, Direct, Inverse, Joint and Combined Variation, Introduction to the Graphing Display Calculator (GDC), Systems of Linear Equations and Word Problems, Algebraic Functions, including Domain and Range, Scatter Plots, Correlation, and Regression, Solving Quadratics by Factoring and Completing the Square, Solving Absolute Value Equations and Inequalities, Solving Radical Equations and Inequalities, Advanced Functions: Compositions, Even and Odd, and Extrema, The Matrix and Solving Systems with Matrices, Rational Functions, Equations and Inequalities, Graphing Rational Functions, including Asymptotes, Graphing and Finding Roots of Polynomial Functions, Solving Systems using Reduced Row Echelon Form, Conics: Circles, Parabolas, Ellipses, and Hyperbolas, Linear and Angular Speeds, Area of Sectors, and Length of Arcs, Law of Sines and Cosines, and Areas of Triangles, Introduction to Calculus and Study Guides, Basic Differentiation Rules: Constant, Power, Product, Quotient and Trig Rules, Equation of the Tangent Line, Tangent Line Approximation, and Rates of Change, Implicit Differentiation and Related Rates, Differentials, Linear Approximation and Error Propagation, Exponential and Logarithmic Differentiation, Derivatives and Integrals of Inverse Trig Functions, Antiderivatives and Indefinite Integration, including Trig Integration, Riemann Sums and Area by Limit Definition, Applications of Integration: Area and Volume, \(\displaystyle {{\log }_{2}}\frac{1}{{16}}=x\), \(\displaystyle {{\log }_{{32}}}\frac{1}{4}=x\), \(y={{\log }_{3}}\left( {2\left( {x-1} \right)} \right)-1\), \(\left( {-\infty ,\,\,\infty } \right)\). A negative sign inside the parentheses means reflecting across the \(y\)–axis. If the same log and same base are on both sides, you can just set arguments of logs equal to each other; this is called the “One-to-One Property of Logarithms”: Solving Logs: Set Arguments of Logs with Same Base Equal to Each Other, \(\begin{align}\ln {{4}^{3}}&=\ln x\\{{4}^{3}}&=x\\x&=64\end{align}\), \(\begin{align}{{\log }_{6}}\left[ {\left( {2x-5} \right)\left( x \right)} \right]&={{\log }_{6}}10\\2{{x}^{2}}-5x&=10\\2{{x}^{2}}-5x-10&=0\\x&=\frac{{5\pm \sqrt{{25-\left( 4 \right)\left( 2 \right)\left( {-10} \right)}}}}{4}=\frac{{5\pm \sqrt{{105}}}}{4}\\x&=\frac{{5+\sqrt{{105}}}}{4}\text{ }\,\,\,\text{ (negative root won }\!\!’\!\!\text{ t work)}\end{align}\).

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