Algebra 2 Logs

Interest

• Compound Interest

  • A=P(1+r/n)^nt

    • n : # of times compounded per year

• Compound Continuously

  • A=Pe^rt

Exponential Logarithms

• b^p=n →log_b(n)=P

  Rules:

• b > 0 (be positive) b not = to 0

Inverse Properties

Ex)

1) log₂64^x

log₂(2⁶)^x=log₂(2²)^x=6x

2) log_3⁵√9

log₃9^1/2=log₃(3²)^1/5=log₃3^2/5=2/5

Properties of Logs

log_b(mn) = log_bm+log_bn

log_b(m/n) = log_bm-log_bn

log_bmⁿ = nlog_bm

Change the Base

log_an = (log_bn / log_ba)

Ex) log₃8

log8/log3 = 1.893

Solving Exponential

Same Base

1. set exponents = to each other & solve

   ex) e^2x=e^3x-1

   → 2x=3x-1 →x=1

Different Base

1. rewrite in log form

2. take log of both sides

   ex) 7^5x - 2=10 →7^5x =12 (turn to log) →log₇12=5x → 1.277=5x → x=0.2554

Solving Logs

Same Base on Logs

1. Condense if needed

2. set x=y

3. Check answer

   ex) log(x-1)+log(x+3) = log (x²-4x)

   →(Condense) log(x-1)(x+3) (FOIL) …x=.5 plug in No Solution

Different log_bx=n

1. Condense log

   a) Write as an exponential

   b) Raise to the base of both sides

   Ex) log₂(5x-17)=3

   A:

   2³=5x-17 → 8=5x-17 → 25=5x → x=5

   B:

   2^{log₂(5x-17)}=2³ →5x-17=8 →5x=25 → x=5

Find the Inverse

1. switch x & y

2. Isolate base/log

3. Solve for y

   NOTE:

   Exponent→Log & Log→Exponent

   Ex) y=6x

   x=6y → log₆x=y

Graphing

	Exponential Function	Log Function
Parent Function	y=bx	y=logbx
Graph
Domain	(-∞,∞)	(0,∞)
Range	(0,∞)	(-∞,∞)
Asymptote	Horizontal y=0	Vertical x=0
Inverse	Is a log! x=by → logbx=y	Is a Exp! logby=x →bx=y
Transformation	y=a*bx-h+k y=k (horizontal asymptote)	y=alogb(x-h)+k x=h (vertical asymptote)

Transformations

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