Warm-up

Solve: log3(x+3) + log32 = 2

log32(x+3) = 2

log3 2x + 6 = 2

32 = 2x + 62x + 6 = 92x = 3x = 3/2

Exponential and Logarithmic Equations

Section 3-4

Objectives

I can solve equations with exponents using logarithms

4

x

y

Graph f (x) = log2 x

Since the logarithm function is the inverse of the exponential function of the same base, its graph is the reflection of the exponential function in the line y = x.

83

42

21

10

–1

–2

2xx

4

1

2

1

y = log2 x

y = xy = 2x

(1, 0)

x-intercept

horizontal asymptote y = 0

vertical asymptote x = 0

5

The function defined by f(x) = loge x = ln x

is called the natural logarithm function.

Use a calculator to evaluate: ln 3, ln –2, ln 100

ln 3

ln –2

ln 100

Function Value Keystrokes Display

LN 3 ENTER 1.0986122

ERRORLN –2 ENTER

LN 100 ENTER 4.6051701

y = ln x

(x 0, e 2.718281)

y

x

5

–5

y = ln x is equivalent to e y = x

6

Log Functions Overview

Log Function Baseloga x a

log x 10loge x e

ln x e

Using Logarithms to Solve ExponentsSo far we have solved exponents using

the principle of getting the same base, then setting the exponents equal.

There are many times that we cannot get the same base, so we need to solve a different method.

Old Problems vs New

Solve for x2(x+1) = 82(x+1) = 23

x+1 = 3x =2

Solve for x8(2x-5) = 5(x+1)

If this problem we cannot get the same base

To work this new type problem, we will use logarithms

Rule for Logarithms

Logarithms can be applied to equations.In any equation, if I do something to one

side, I must do the same thing to the other side to keep equality.

In these problems, we will take the Common Log or Natural log of both sides of each equation, then use the Power Property

Solve for x

8(2x-5) = 5(x+1)

log 8(2x-5) = log 5(x+1)

(2x-5) log 8 = (x+1) log 5(2x–5) (.9031) = (x+1) (.6990)1.81x – 4.52 = .699x + .6991.11x = 5.22x = 4.70

Natural log vs ex

Use your calculator and determine the following:

Ln e1 = Ln e2 = Ln e3 =

ln xe xln xe x

12

3

Example 2

e3x = 20ln e3x = ln 203x = ln 203x = 2.9957x = .9986

Example 3

3 + ln x = -8ln x = -11eln x = e-11

x = 1.67 x 10-5

3)7(loglog 22 xx

3)7(log2 xx32)7( xx

872 xx

0872 xx0)1)(8( xx

18 xorx

Homework

WS 6-4