2 Section Outline The Natural Logarithm of x Properties of the Natural LogarithmExponential ExpressionsSolving Exponential EquationsSolving Logarithmic EquationsOther Exponential and Logarithmic FunctionsCommon LogarithmsMax’s and Min’s of Exponential Equations
3 The Natural Logarithm of x DefinitionExampleNatural logarithm of x: Given the graph of y = ex, the reflection of that graph about the line y = x, denoted y = ln x
5 Properties of the Natural Logarithm The point (1, 0) is on the graph of y = ln x [because (0, 1) is on the graph of y = ex].ln x is defined only for positive values of x.ln x is negative for x between 0 and 1.ln x is positive for x greater than 1.ln x is an increasing function and concave down.
6 Exponential Expressions EXAMPLESimplify.SOLUTIONUsing properties of the exponential function, we have
7 Solving Exponential Equations EXAMPLESolve the equation for x.SOLUTIONThis is the given equation.Remove the parentheses.Combine the exponential expressions.Add.Take the logarithm of both sides.Simplify.Finish solving for x.
8 Solving Logarithmic Equations EXAMPLESolve the equation for x.SOLUTIONThis is the given equation.Divide both sides by 5.Rewrite in exponential form.Divide both sides by 2.
10 Common Logarithms Definition Example Common logarithm: Logarithms to the base 10
11 Max’s & Min’s of Exponential Equations EXAMPLEThe graph of is shown in the figure below. Find the coordinates of the maximum and minimum points.
12 Max’s & Min’s of Exponential Equations CONTINUEDAt the maximum and minimum points, the graph will have a slope of zero. Therefore, we must determine for what values of x the first derivative is zero.This is the given function.Differentiate using the product rule.Finish differentiating.Factor.Set the derivative equal to 0.Set each factor equal to 0.Simplify.
13 Max’s & Min’s of Exponential Equations CONTINUEDTherefore, the slope of the function is 0 when x = 1 or x = -1. By looking at the graph, we can see that the relative maximum will occur when x = -1 and that the relative minimum will occur when x = 1.Now we need only determine the corresponding y-coordinates.Therefore, the relative maximum is at (-1, 0.472) and the relative minimum is at (1, -1).