#14 Exponentials and Logs

What is an exponential function?

A function in the form f(x) = a^x

Where does an exponential cross the y-axis?

y = 1

If a>1, which way does the graph tend?

Upwards

If 0<a<1, which way does the graph tend?

Downwards

If f(x) = e^kx, what is f ‘(x) ?

ke^kx

What is the constant e used for?

To model population growth where the rate of increase is proportional to the size of the population

What is a logarithm?

The inverse of an exponential function

State the expression log_an = x as an exponential?

a^x = n

State the multiplication law of logarithms.

log_ax + log_ay = log_axy

State the division law of logarithms.

log_ax - log_ay = log_ax/y

State the power law of logarithms.

log_a(x^k) = klog_ax

State the special cases of log laws.

log_a(1/x) = -log_ax

log_aa = 1

log_a1 = 0

Describe how to use logs to solve equations in the form a^x = b.

Re-arrange into log form and solve using a calculator

Describe how to solve an equation involving hidden quadratics.

Use the substitution method to make a quadratic and solve, subbing the answers into the substitution equation

State how to solve more complex equations using logs.

Take logs of both sides

What is the natural log?

The reverse function of e^x

How can a linear relationship be modelled from data with the equation y=axⁿ?

logy = loga + nlogx

compare the y = mx + c

How can a linear relationship be modelled from data with the equation y=ab^x?

logy = loga + xlogb

compare to y = mx + c