Logarithms Flashcards
What is a logarithm
related to exponentials, used to solve exponential equations
2^3 = 8 =…
log2 8 = 3
Restriction on log function
base a>1
base a cannot = 0
if a^x = y…
loga y = x
Graph of y=a^x characteristics
a^x always positive, y-int: (0,1) Domain: (-8, 8) Range: (0, 8) Asymptote: y=0
Graph of y=a^-x
Reflection of y=a^x about the y-axis
same y-int, domain, range, asymptote
When a = 1/2…
= 2^-x
Graph of y=loga x characteristics
x-int: (1,0)
domain: (0,8)
range: (-8, 8)
asymptote: x=0
loga (mn)=…
loga m + loga n
loga (m/n)=…
loga m - loga n
loga (m^n)=…
n loga m
loga 1=…
0
loga a=…
1
loga a^n=…
n
a^loga x=…
x
change of base law
loga x = logb x/logb a
x on top, base on bottom
How to find derivative of exponential function using first principles
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
what is Euler’s number
e^x
the number that gives exactly the same graph for a derivative function
derivative of e^x
e^x
derivative of ke^x
ke^x
How to differentiate y=ke^f(x)
Use chain rule (dy/dx = dy/du x du/dx) (let u = f(x))
How to differentiate y=ke^f(x) fast way
Bring down derivative of power and multiply
power stays the same
what is the natural logarithm
logarithm to base e loge x (ln x)
y=e^x and y=loge x are…
symmetrical about the line y=x
for inverse functions
the x and y values are interchanged
Why do you always have to check for valid solutions when solving log equations
because there is restrictions
a > 0, a cannot = 1, and x > 1
Why do we need logs to solve equations
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
How to solve equations with logs
- Rewrite in log form and use change of base rule
- Take the log of both sides, then use log laws
(don’t forget to check if a log is negative when working with inequalities)
Natural log laws
loge e^x = x
e^loge e^x = x
If both sides have the same base…
Solve the powers in a equation for x
How to solve equations with e in it
Take the log base e of each side (not log 10)
Graph of y=a^x as x changes
x<0, graph approaches x-axis faster
x>0 graph approaches y-axis faster (steeper)
graph of y = ka^x
dilates the graph of y=a^x by a factor of k
graph of y = ka^x + c
Shifts graph of y = ka^x up (+) or down (-)
graph of y = ka^x+b
Shifts the graph of y = ka^x left (+) or right (-)
graph of y=loga x as base a changes
as a increases:
01, graph approaches x-axis faster
graph of y = klog a x
Dilates graph by factor of k (enlarged)
times all coordinated by factor k
graph of y = kloga x + c
Shifts graph up (+) or down (-)
graph of y = kloga (x+b)
Shifts graph left (+) or right (-)
When drawing a graph from a log question…
x-axis - letter on the right
y-axis - letter on the left
How to find equation of exponential given the graph
- Start with general equation y=a^x
- Look at asymptote to see if shifted up or down
- Sub a point given then solve to find a
How to find equation of log given the graph
- Start with general equation y=loga x
- Look at asymptote to see if shifted up or down
- Sub point into equation to find a
How to find the rate of change with logs/exponentials
find the derivative of the original equation then sub in value for t
