Sections 5.3-5.5

Logarithmic Functions (5.3)

What is a logarithm???LOGS ARE POWERS!!!!A logarithm or “log” of a number of a certain base is the exponent to which the base of the log must be raised in order to produce the number. The base cannot equal 1 and must be greater than 0.

For instance, if logb(x) = c and b≠1 and b>0, then c is the specific exponent to which you must raise b in order to get x: bc = x

Logarithmic FunctionsWhy do we need logs? Let’s explore…

32 = 9 and 33 = 27

but what would we need to raise 3 to in order to get 20??

3a = 20

that’s what logs tell us!!

a = log320

Which two integers is log320 between?

2 and 3

Logarithmic Functions

From the definition, we have stated that if logb(x) = c, then bc = x under the conditions that b≠1 and b>0.

Why do we need to place any restrictions on b or x so this can make sense? Let’s try some values…

logb(x) = c, so bc = xlog2(8) = c so 2c = 8

c = 3, so far we are ok

log1(5) = c so 1c = 5

Does not exist; 1c always equals 1

log-2(8) = c so (-2)c = 8

Does not exist; if c = 3, then (-2)3 = -8

log3(-9) = c so 3c = -9

Does not exist; 3c cannot be negative

log2(0) = c so 2c = 0

Does not exist; 2c cannot equal 0

Summary: b≠1, b>0 and x>0

Log Properties

(1) logbb = 1

(2) logb1 = 0

common log has base 10: log(x) = log10(x)

natural log has base e: ln(x) = loge(x)

Therefore…

(1) log10 = 1

(2) lne = 1

Practice

Evaluate, if possible. If not, state so.

y = logb(x - h) + k

When graphing logs we first need to identify and graph the asymptote. Earlier we discovered that the argument inside the log must be greater than 0.

Therefore, x > h so the domain is (h, +∞) and there must be an asymptote at x = h

The range is all real numbers

Now find three points; the simplest values are when x - h = 1 and when x - h = b

Graph of a Logarithmic Function

Graph of y = logb(x) when b>1

Can you state any characteristics?–Asymptotes, x - intercepts, Domain, Range

• Graph of y = logb(x) when 0<b<1

Practice

Graph. State the domain and range.

Change of Base Formula

This formula allows us to compute logs using the calculator, by converting to base 10 or e.

Example: