14. Exponentials And Logarithms Flashcards
Exponential graphs when a > 1
Cut the y axis at 1
Tend towards 0 when x tends towards negative infinity
Tend towards infinity when x tends towards infinity
If a is smaller in y = a^x
When x < 0: y values are higher
When x = 0: y values are equal (1)
When x > 0: y values are lower
Transformation from y = a^x to y = (1/a)^x
Reflection in the y-axis
Transformations of exponential graphs
Work the same way as if x wasn’t a power
The exponential function
y = e^x
What is so special about the exponential function
The differentiated function is still the same
y = e^kx differentiated
ke^kx
loga n
The a is in subscript
a^x = n
Finds what power you need to raise a by to get n
What can’t you log on?
Negatives, but it can output them
log(ab)
OPTN, F4, F6, F4
base, output
ln
“natural log of e”
Uses e as the base
calculator button log
Always uses 10 as a base
log a x + log a y
log a (xy)
log a x - log a y
log a (x/y)
log a x^k
k log(a) x
3^x = 20 as a log
x = log 3 20
If you have two different bases with different powers
Write each as a log without a base
ln e^x
x
e^lnx
x
How to solve e^2x + a e^x + b = 0
Let y = e^x and solve as a quadratic before using ln
Remember e^2x = (e^x)^2)
What to do if you have e^-x
Multiply every value by e^x to make a hidden quadratic with the e^-x becoming the integer
e^x = y
ln(e^x) = ln(y) x = ln(y)
ln (x) = y
e ^(ln(x)) = e^y
x = e^y
Polynomial -> linear
y = ax^n log y = log ax^n log y = log x^n + log a log y = n log x + log a Compare against y = mx + c (gradient n, y-intercept log a)
Exponential -> linear
y = ab^x
log y = log a + x log b
Compare against y = mx + c (gradient log b, y-intercept log a)
Graph of log y = … to a n^x
10^y-intercept = a 10^gradient = n
3^(x+1)
3 x 3^x
ln (x) = 2
x = e^2
Logarithmic equations where both are to a power of ax + y
Take the log/ln of both sides Put the power in front of the log/ln Separate the x and integer coefficient on each side Rearrange to get all x on one side Factor out x and divide to find x
