Log Review

Rewriting

100

log_b(x)=y

b^y=x

200

ln(x)=y

e^y=x

300

log_x13=7

x⁷=13

400

ln(43)=x

e^x=43

500

log₂5³=x

2^x=5³

Change of Base Formula

100

What is the change of base formula?

log_bw=log(w)/log(b)

200

Using C.o.B. Rewrite the following: 

log_qk=___________

log_qk=log(k)/log(q)

300

log₇29=_________ 

Rewrite with a base of 10.

log₇29 = log(29)/log(7) = 1.73

400

log₁₂(1.9)=_________ 

Rewrite with a base of 10.

log₁₂(1.9) = .26

500

Rewrite with a base of 10.

log₃5³ = ______

log₃5³ = 4.39

Expansion Formulas

100

Multiplication Expansion Rule

log(a*b) = ___________

log(a*b) = log(a) + log(b)

200

Division Expansion Rule

log(a/b)=_______

log(a/b) = log(a) - log(b)

300

log(b^m) = _____________

log(b^m) = m * log(b)

400

ln((a*b)/c) = __________

ln((a*b)/c) = ln(a*b) - ln(c) = ln(a) + ln(b) - ln(c)

500

log(j/z)ⁿ = ____________

log(j/z)ⁿ = n*log(j/z) = n*(log(j) - log(z))

or          n*log(j) - n* log(z)

Solving Log & LN Problems

100

Solve for x:

log(x) + 2 = 5

x = 10³ = 1000

200

Solve for x:

log(5x) + 9 = 11

x = 20

300

Solve for x:

log(-x+7)⁵ = -15

x = -10^-3 + 7 = 6.99

400

Solve for x:

4 * log(x+4)² = 16

x = 96

500

Solve for x:

-5 * log₃(3x+2)³ = -75

x=80.33

Solving Exponential Problems

100

Solve for a

3^a=69

a=3.85

200

Solve for x:

e^x=47

x=3.85

300

Solve for h:

9^h+10+3=81

h=-8.017

400

Solve for b:

-10e^2-2b-6=-66

b=0.104

500

Solve for n:

6*16^n-7+5=35

n=7.58