2Section 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
3The 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
5Properties 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.
6Exponential Expressions EXAMPLESimplify.SOLUTIONUsing properties of the exponential function, we have
7Solving 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.
8Solving 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.
10Common Logarithms Definition Example Common logarithm: Logarithms to the base 10
11Max’s & Min’s of Exponential Equations EXAMPLEThe graph of is shown in the figure below. Find the coordinates of the maximum and minimum points.
12Max’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.
13Max’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).