Chapter 14 Exponentials and Logarithms Flashcards
What are exponential functions?
Functions of the form f(x) = a^x
If f(x) = e^x what is f’(x)?
f’(x) = e^x
If f(x) = e^kx what is f’(x)?
f’(x) = ke^kx
What can e^x be used to model?
Situations where the rate of increase is proportional to the size of the population at any given moment
What can e^-x be used to model?
Situations where the rate of decrease is proportional to the size of the population remaining at any given moment
What is loga(n) = x equivalent to?
a^x = n
What is a natural logarithm and what is the symbol for it?
A logarithm to the base e
Symbol: ln
What is the multiplication law for logarithms?
loga(x) + loga(y) = loga(xy)
What is the division law for logarithms?
loga(x) - loga(y) = loga(x/y)
What is the power law for logarithms?
loga(x^k) = kloga(x)
What is the power law for logarithms when k = -1?
loga(x^-1) = -loga(x)
What is loga(a) equivalent to when a>0 and a doesn’t equal 1?
loga(a) = 1
What is loga(1) equivalent to when a>0 and a doesn’t equal 1?
loga(1) = 0
What is the relationship between the graph of y = ln x and the graph y = e^x?
The graph of y = ln x is a reflection of the graph y = e^x in the line y = x
What is e^ln x equal to?
e^(ln x) = ln (e^x) = x ln(e) = x
How do you convert y = ax^n into the form y = mx+c?
- Take logs of both sides: log y = log ax^n
- Use the multiplication law: log y = log a + log x^n
- Use the power law: log y = log a + nlog x
This is in the form y = mx + c with n being the gradient and c being log a
How do you convert y = ab^x into the form y = mx+c?
- Take logs of both sides: log y = logab^x
- Use the multiplication law: log y = log a + log b^x
- Use the power law: log y = log a + xlog b
This is in the form y = mx + c with log b being the gradient and c being log a
What is log a equivalent to?
log10(a)
If f(x) = g(x) what is loga f(x) equal to?
loga g(x)
