Aim: Logarithm Equations Course: Alg. 2 & Trig. Aim: How do we solve logarithm equations? Do Now:
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Transcript of Aim: Logarithm Equations Course: Alg. 2 & Trig. Aim: How do we solve logarithm equations? Do Now:
Aim: Logarithm Equations Course: Alg. 2 & Trig.
Aim: How do we solve logarithm equations?
Do Now:
Aim: Logarithm Equations Course: Alg. 2 & Trig.
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
Aim: Logarithm Equations Course: Alg. 2 & Trig.
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
Aim: Logarithm Equations Course: Alg. 2 & Trig.
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
Aim: Logarithm Equations Course: Alg. 2 & Trig.
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
Aim: Logarithm Equations Course: Alg. 2 & Trig.
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
Aim: Logarithm Equations Course: Alg. 2 & Trig.
Model Problem
x2x + 2x = 0
3 435 70 x ln( / )e