Chapter 14 - Exponentials and logarithms

Question 1

If f(x) = e^x, what is f’(x)?

If y = e^x, what is dy/dx?

Answer 1

f’(x) = e^x

dy/dx = e^x

Question 2

If f(x) = e^kx, what is f’(x)?

If y = e^kx, what is dy/dx?

Answer 2

f’(x) = ke^kx

dy/dx = ke^kx

Question 3

What does log_an mean?

Answer 3

What power do I need to raise “a” by to get to “n”?

a^x = n

Question 4

How do I write a^x = n in log format?

Answer 4

log_an = x

Question 5

What is log_ax + log_ay equal to?

Answer 5

log_a (xy)

Question 6

What is log_ax - log_ay equal to?

Answer 6

log_a(x/y)

Question 7

What is log_a (x^k) equal to?

Answer 7

k log_a x

Question 8

What is log_a (1/x) equal to?

Answer 8

log_a (x^-1) = - log_a x

Question 9

What is log_a a?

Answer 9

1

Question 10

What is log_a 1?

Answer 10

0

a⁰ = 1

Question 11

What is log_a (any negative number)?

Answer 11

Not defined, you cannot find the log of a negative number.

Question 12

If f(x) = g(x) what do you get if you “take logs of each side?”

Answer 12

log_a f(x) = log_a g(x)

Question 13

If you reflect y = ln x in the line y = x, waht do you get?

Answer 13

y = e^x

Question 14

What does e^{ln x} equal?

Answer 14

ln (e^x) = x

Question 15

What does ln (e^x) equal?

Answer 15

e^{ln x} = x

Question 16

If y = axⁿ, what do you get if you take logs each side?

Answer 16

log y = log (axⁿ)

rewrite log (axⁿ) as log a + n log x

so, log y = n log x + log a

straight line of log y (vertical axis) against log x (horizontal axis). gradient n and vertical intercept of log a.

Question 17

If y = ab^x, what do you get if you take logs of each side?

Answer 17

log y = log (ab^x)

rewrite log (ab^x) as log a + x log b

so, log y = x log b + log a

straight line of log y (vertical axis) against x (horizontal axis) with gradient log b and vertical intercept log a