Warm-up

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Warm-up

• Solve log3(x3) log32 2
• log32(x3) 2
• log3 2x 6 2
• 32 2x 6
• 2x 6 9
• 2x 3
• x 3/2

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Exponential and Logarithmic Equations Section
3-4
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Objectives

• I can solve equations with exponents using
  logarithms

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Graph f(x) log2 x
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.
y 2x
y x
y log2 x
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Natural Logarithmic Function
y ln x
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 3 ENTER
1.0986122
ln 2
ERROR
LN 2 ENTER
ln 100
LN 100 ENTER
4.6051701
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Log Functions Overview

• Log Function Base
• loga x a
• log x 10
• loge x e
• ln x e

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Using Logarithms to Solve Exponents

• So 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.

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Old Problems vs New

• Solve for x
• 2(x1) 8
• 2(x1) 23
• x1 3
• x 2

• Solve for x
• 8(2x-5) 5(x1)
• If this problem we cannot get the same base
• To work this new type problem, we will use
  logarithms

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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

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Solve for x

• 8(2x-5) 5(x1)
• log 8(2x-5) log 5(x1)
• (2x-5) log 8 (x1) log 5
• (2x5) (.9031) (x1) (.6990)
• 1.81x 4.52 .699x .699
• 1.11x 5.22
• x 4.70

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Natural log vs ex

• Use your calculator and determine the following
• Ln e1
• Ln e2
• Ln e3

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Example 2

• e3x 20
• ln e3x ln 20
• 3x ln 20
• 3x 2.9957
• x .9986

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Example 3

• 3 ln x -8
• ln x -11
• eln x e-11
• x 1.67 x 10-5

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Homework

• WS 6-4