Logarithms

Question 1

What is a logarithm

Answer 1

related to exponentials, used to solve exponential equations

Question 2

2^3 = 8 =…

Answer 2

log2 8 = 3

Question 3

Restriction on log function

Answer 3

base a>1

base a cannot = 0

Question 4

if a^x = y…

Answer 4

loga y = x

Question 5

Graph of y=a^x characteristics

Answer 5

a^x always positive,
y-int: (0,1)
Domain: (-8, 8)
Range: (0, 8)
Asymptote: y=0

Question 6

Graph of y=a^-x

Answer 6

Reflection of y=a^x about the y-axis

same y-int, domain, range, asymptote

Question 7

When a = 1/2…

Answer 7

= 2^-x

Question 8

Graph of y=loga x characteristics

Answer 8

x-int: (1,0)

domain: (0,8)
range: (-8, 8)
asymptote: x=0

Question 9

loga (mn)=…

Answer 9

loga m + loga n

Question 10

loga (m/n)=…

Answer 10

loga m - loga n

Question 11

loga (m^n)=…

Answer 11

n loga m

Question 12

loga 1=…

Answer 12

0

Question 13

loga a=…

Answer 13

1

Question 14

loga a^n=…

Answer 14

n

Question 15

a^loga x=…

Answer 15

x

Question 16

change of base law

Answer 16

loga x = logb x/logb a

x on top, base on bottom

Question 17

How to find derivative of exponential function using first principles

Answer 17

same way as usual (lim h–> 0 f(x+h) -f(x)/h)

but substitute a small value of x into h instead of 0 and put x value in front

f(x) = a^x
f'(x) = a^x lim h->0 a^h -1/h

Question 18

what is Euler’s number

Answer 18

e^x

the number that gives exactly the same graph for a derivative function

Question 19

derivative of e^x

Answer 19

e^x

Question 20

derivative of ke^x

Answer 20

ke^x

Question 21

How to differentiate y=ke^f(x)

Answer 21

Use chain rule (dy/dx = dy/du x du/dx)
(let u = f(x))

Question 22

How to differentiate y=ke^f(x) fast way

Answer 22

Bring down derivative of power and multiply

power stays the same

Question 23

what is the natural logarithm

Answer 23

logarithm to base e
loge x (ln x)

Question 24

y=e^x and y=loge x are…

Answer 24

symmetrical about the line y=x

Question 25

for inverse functions

Answer 25

the x and y values are interchanged

Question 26

Why do you always have to check for valid solutions when solving log equations

Answer 26

because there is restrictions | a > 0, a cannot = 1, and x > 1

Question 27

Why do we need logs to solve equations

Answer 27

Previously we solved equations by changing the base to the same and equating indicies (2^x+1 = 16 2^x+1 = 2^4 x+1=4, x=3) but we can only do that when the base can be changed, so we use logs to solve these equations

Question 28

How to solve equations with logs

Answer 28

1. Rewrite in log form and use change of base rule 2. Take the log of both sides, then use log laws (don't forget to check if a log is negative when working with inequalities)

Question 29

Natural log laws

Answer 29

loge e^x = x | e^loge e^x = x

Question 30

If both sides have the same base...

Answer 30

Solve the powers in a equation for x

Question 31

How to solve equations with e in it

Answer 31

Take the log base e of each side (not log 10)

Question 32

Graph of y=a^x as x changes

Answer 32

x<0, graph approaches x-axis faster | x>0 graph approaches y-axis faster (steeper)

Question 33

graph of y = ka^x

Answer 33

dilates the graph of y=a^x by a factor of k

Question 34

graph of y = ka^x + c

Answer 34

Shifts graph of y = ka^x up (+) or down (-)

Question 35

graph of y = ka^x+b

Answer 35

Shifts the graph of y = ka^x left (+) or right (-)

Question 36

graph of y=loga x as base a changes

Answer 36

as a increases: | 01, graph approaches x-axis faster

Question 37

graph of y = klog a x

Answer 37

Dilates graph by factor of k (enlarged) | times all coordinated by factor k

Question 38

graph of y = kloga x + c

Answer 38

Shifts graph up (+) or down (-)

Question 39

graph of y = kloga (x+b)

Answer 39

Shifts graph left (+) or right (-)

Question 40

When drawing a graph from a log question...

Answer 40

x-axis - letter on the right | y-axis - letter on the left

Question 41

How to find equation of exponential given the graph

Answer 41

1. Start with general equation y=a^x 2. Look at asymptote to see if shifted up or down 3. Sub a point given then solve to find a

Question 42

How to find equation of log given the graph

Answer 42

1. Start with general equation y=loga x 2. Look at asymptote to see if shifted up or down 3. Sub point into equation to find a

Question 43

How to find the rate of change with logs/exponentials

Answer 43

find the derivative of the original equation then sub in value for t