Log Properties

Question 1

Natural base e

Answer 1

y=e^x exponential function

y=e^-x (inverse)

Question 2

f(x) = b^x vs f(x) = e^x

Answer 2

Growth: b > 0 vs x >0
Decay: 0

Question 3

Which 2 functions are the same?
y=2^x
y=2^-x
y=(1/2)^x

Answer 3

y=2^-x
y=(1/2)^x
Bc y=2^-x – 1/2^x – (1/2)^x

Question 4

Natural Base Functions

y=ae^rx

Answer 4

Growth / decay depends on whether the x is negative or positive

Question 5

Continuously compounded interest

Answer 5

A =Pe^rt
y=5e^0.25x – growth
y=0.8e^-3x – decay

Question 6

An exponential function is the inverse of a logarithmic function

Answer 6

f(x)=b^x — g(x)=log_b_ x

b means subscript b

Question 7

b^x=m —

Answer 7

log_b_m = x

Ex: 2^x = 8 — log_2_ 8 = 3

Question 8

y=2^x and x=2^y are

Answer 8

Inverses

Question 9

x=2^y and y=log_2_x are

Answer 9

The same thing

Question 10

Common log

Answer 10

y=logX

base 10

Question 11

Exponential vs Logarithmic graph

Answer 11

Exponential:
g(x)=b^x, (0,1), asymptote = x-axis, domain: all reals, range: y>0
Logarithmic:
g(x)=log_b_x, (1,0), asymptote = y-axis, domain: x>0, range: all reals

Question 12

log_b_Y=x if

Answer 12

b^x=Y

Question 13

Base cancellation

Answer 13

common bases cancel out

lne^x^3 — log_e_e^x^3 (both e’s cancel out) =x^3

Question 14

ln = natural log

Answer 14

ln = log_e_
log_e_X = lnX

Question 15

log_b_mn =

Answer 15

log_b_m + log_b_n

Question 16

log_b_ m/n =

Answer 16

log_b_m - log_b_n

Question 17

log_b_m^n =

Answer 17

nlog_b_m

Question 18

log_b_b =

Answer 18

1

Question 19

log_b_1 =

Answer 19

0

Question 20

logX =

Answer 20

log_10_X

Question 21

lne^2 + lne^5 =

Answer 21

common base
ln(e^2 * e^5)
log_e_e^7
=7

Question 22

log_3_7 =

Answer 22

log7 / log3 = ln7 / ln3

Question 23

x=log_e_(y-4) —

Answer 23

e^x = y-4