Just as addition is the inverse of subtraction and multiplication and division are inverses of one another, logarithms and exponents are inverse operations.

## Definition of a Logarithmic Function with Base a

A logarithmic function is defined as y = log_{a }x if and only if a^{y} = x

For x > 0 and a > 0, and a ≠1

That means the answer to a logarithmic equation is the exponent of an exponential equation.

How would we write the following in exponential form?

Log_{10 }100 = 2

Log_{3} 81 = 4

### Write the following in logarithmic form.

2^{8}= 256

5^{3} = 125

17^{0} = 1

## Properties of Logarithms

log_{a} 1 = ________ because

log_{a} a = ________ because

log _{a} a^{x} = ________ because

If log _{a} x = log _{a} y then

### Solve the following:

log _{4} 16 =

log _{6} 1 =

log _{9} 3 =

log _{2} (-1) =

### Graph y = log _{2} x

Start by writing it in exponential form.

2^{y} = x

Now, plug in values for **y**.

This should remind you of a function you have been introduced to earlier:

f(x) = 2^{x}

How are the two related?

Graph y = log _{4} (x – 3)

Remember your families of graphs? Start by graphing your parent function of y = log _{4} x. Then shift it 3 to the right.

## The Natural Log

e – Euler’s Number, Napier’s Constant, the Natural Base

e ≈ 2.718281

e is the base in the natural log noted as ln.

So ln = log_{e} ≈ log _{2.718}

Graph y = ln x

Graph y = -ln (x – 1)

Because order of operations says to do what’s in parenthesis first, shift left one first, then reflect across the x-axis.

If your logarithmic function involves a reflection and a vertical shift like this function – y = -ln x + 2, first reflect over the x – axis, then shift up 2 because the multiplication of the -1 would take place before the addition of 2.

Solve log _{12} (x – 3) = log _{12} (2x + 5)

Precalculus with Limits – Larson & Hostetler p. 229-238