14. Exponentials And Logarithms

Question 1

Exponential graphs when a > 1

Answer 1

Cut the y axis at 1
Tend towards 0 when x tends towards negative infinity
Tend towards infinity when x tends towards infinity

Question 2

If a is smaller in y = a^x

Answer 2

When x < 0: y values are higher
When x = 0: y values are equal (1)
When x > 0: y values are lower

Question 3

Transformation from y = a^x to y = (1/a)^x

Answer 3

Reflection in the y-axis

Question 4

Transformations of exponential graphs

Answer 4

Work the same way as if x wasn’t a power

Question 5

The exponential function

Answer 5

y = e^x

Question 6

What is so special about the exponential function

Answer 6

The differentiated function is still the same

Question 7

y = e^kx differentiated

Answer 7

ke^kx

Question 8

loga n

Answer 8

The a is in subscript
a^x = n
Finds what power you need to raise a by to get n

Question 9

What can’t you log on?

Answer 9

Negatives, but it can output them

Question 10

log(ab)

Answer 10

OPTN, F4, F6, F4

base, output

Question 11

ln

Answer 11

“natural log of e”

Uses e as the base

Question 12

calculator button log

Answer 12

Always uses 10 as a base

Question 13

log a x + log a y

Answer 13

log a (xy)

Question 14

log a x - log a y

Answer 14

log a (x/y)

Question 15

log a x^k

Answer 15

k log(a) x

Question 16

3^x = 20 as a log

Answer 16

x = log 3 20

Question 17

If you have two different bases with different powers

Answer 17

Write each as a log without a base

Question 18

ln e^x

Answer 18

x

Question 19

e^lnx

Answer 19

x

Question 20

How to solve e^2x + a e^x + b = 0

Answer 20

Let y = e^x and solve as a quadratic before using ln

Remember e^2x = (e^x)^2)

Question 21

What to do if you have e^-x

Answer 21

Multiply every value by e^x to make a hidden quadratic with the e^-x becoming the integer

Question 22

e^x = y

Answer 22

ln(e^x) = ln(y)
x = ln(y)

Question 23

ln (x) = y

Answer 23

e ^(ln(x)) = e^y

x = e^y

Question 24

Polynomial -> linear

Answer 24

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)

Question 25

Exponential -> linear

Answer 25

y = ab^x
log y = log a + x log b
Compare against y = mx + c (gradient log b, y-intercept log a)

Question 26

Graph of log y = … to a n^x

Answer 26

10^y-intercept = a
10^gradient = n

Question 27

3^(x+1)

Answer 27

3 x 3^x

Question 28

ln (x) = 2

Answer 28

x = e^2

Question 29

Logarithmic equations where both are to a power of ax + y

Answer 29

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