aim: logarithm equations course: alg. 2 & trig. aim: how do we solve logarithm equations? do...

Aim: Logarithm Equations Course: Alg. 2 & Trig.

Aim: How do we solve logarithm equations?

Do Now:


Properties of Logarithms

For any positive numbers M, N, and b, b 1,Each of the following statements is true.

logb MN = logb M + logb N Product Property

logb M/N = logb M – logb N Quotient Property

logb Mk = k logb M Power Property

log (3 • 5) = log 3 + log 5

log (3 / 5) = log 3 – log 5

log 35 = 5 log 3

Note: loga(M + N) ≠ loga M + loga N

Note: base must be the same


x

27

Property of Equality for Log functions

Solving Log Equations using Properties

1. If log A = log B, then A = B:Property of Equality for Log functions

2. Like regular equations use the inverse operation to simplify an equation

Ex. log x - 1/3 log 8 = log 7

x = 14

log x - log 81/3 = log 7Undo Power Law of Logarithms

logx

2log 7

Undo Quotient Law of Logarithms

3. What you do to one side, do exactly to the other


Solving Log Equations using Properties

Ex. log4(x – 3) + log4(x + 3) = 2

log4[( x – 3)(x + 3)] = 2Undo Product Law of Logarithms

When only some terms are logarithmic,consolidate to one side in form logb = N and convert to exponential equation.

Write in Exponential Form( x – 3)(x + 3) = 42

Multiplyx2 – 9 = 16

Check: log4(5 – 3) + log4(5 + 3) = 2

log4(-5 – 3) + log4(-5 + 3) = 2-8

Log4(-8) is undefined; +5 is only answer

x = 5 x = 5


Log Equation Problem

log x + log(x – 3) = 1

log x(x – 3) = 1Undo Product Law of Logarithms

Write in Exponential Formx (x + 3) = 101

Multiplyx2 + 3x = 10

Log of negative number is undefined

Put in Standard Quadratic Formx2 + 3x – 10 = 0

x = -5 x = 2Solve for x

Factor trinomial(x + 5)(x – 2) = 0


Application Problem

The pH of a substance is the concentration of hydrogen ions, [H+], measured in moles of hydrogen per liter of substance. It is given by the formula

Find the amount of hydrogen in a liter of acidrain that has a pH of 4.2.

pH log10

1

[H ]

4.2 log10

1

[H ]

4.2 = log10 1 – log10 [H+]

10-4.2 = [H+]

4.2 = 0 – log10 [H+]

4.2 = – log10 [H+]

– 4.2 = log10 [H+]

log101 = 0

or 0.000063 molesof hydrogen

Always check your answer


Model Problem

x2x + 2x = 0

3 435 70 x ln( / )e

aim: logarithm equations course: alg. 2 & trig. aim: how do we solve logarithm equations? do...

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