Lesson 5-5

Logarithms

Logarithmic functions

Logarithmic functionsThe inverse of

the exponential function.

Logarithmic functionsThe inverse of

the exponential function.

Basic exponential function: f(x) = bx

Logarithmic functionsThe inverse of

the exponential function.

Basic exponential function: f(x) = bx

Logarithmic functionsThe inverse of

the exponential function.

Basic logarithmic function: f-1(x) = logbx

Logarithmic functionsThe inverse of

the exponential function.

Basic logarithmic function: f-1(x) = logbx

Logarithmic functionsThe inverse of

the exponential function.

Basic logarithmic function: f-1(x) = logbx

Every (x,y) (y,x)

Logarithmic functionsBasic rule for changing exponential equations to logarithmic equations

(or vice-versa):

Logarithmic functionsBasic rule for changing exponential equations to logarithmic equations

(or vice-versa):

logbx = a ba = x

Logarithmic functionsBasic rule for changing exponential equations to logarithmic equations

(or vice-versa):

logbx = a ba = x

The base of the logarithmic form becomes the base of the

exponential form.

Logarithmic functionsBasic rule for changing exponential equations to logarithmic equations

(or vice-versa):

logbx = a ba = x

The answer to the log statement becomes the power

in the exponential form.

Logarithmic functionsBasic rule for changing exponential equations to logarithmic equations

(or vice-versa):

logbx = a ba = x

The number you are to take the log of in the log

form, becomes the answer in the exponential form.

Examples:

Examples: log525 = 2 because 52 = 25

Examples: log525 = 2 because 52 = 25

log5125 = 3 because 53 = 125

Examples: log525 = 2 because 52 = 25

log5125 = 3 because 53 = 125

log2(1/8) = - 3 because 2-3 = 1/8

base b exponential function f(x) = bx

base b exponential function f(x) = bx

Domain: All realsRange: All positive reals

base b logarithmic function f-1(x) = logb(x)

base b logarithmic function f-1(x) = logb(x)

Domain: All positive realsRange: All reals

Types of Logarithms

Types of LogarithmsThere are two special logarithms

that your calculator is programmed for:

Types of LogarithmsThere are two special logarithms

that your calculator is programmed for:

log10(x) called the common logarithm

Types of LogarithmsThere are two special logarithms

that your calculator is programmed for:

log10(x) called the common logarithm

For the common logarithm we do not include the

subscript 10, so all you will see is: log (x)

Types of LogarithmsThere are two special logarithms

that your calculator is programmed for:

So, log10(x) log (x) = k if 10k = x

Types of LogarithmsThere are two special logarithms

that your calculator is programmed for:

loge(x) called the natural logarithm

Types of LogarithmsThere are two special logarithms

that your calculator is programmed for:

loge(x) called the natural logarithm

For the natural logarithm, we do not include the subscript

e, so all you will see is: ln (x)

Types of LogarithmsThere are two special logarithms

that your calculator is programmed for:

So, loge(x) ln (x) = k if ek = x

Examples:

Examples:log 6.3 = 0.8 because 100.8 = 6.3

Examples:log 6.3 = 0.8 because 100.8 = 6.3

ln 5 = 1.6 because e1.6 = 5

Example:

Example:Find the value of x to the nearest hundredth.

Example:Find the value of x to the nearest hundredth.

Example:Find the value of x to the nearest hundredth.

10x = 75

Example:Find the value of x to the nearest hundredth.

10x = 75

This transfers to the log statement log 10 75 = x

and the calculator will tell you x = 1.88

Example:Find the value of x to the nearest hundredth.

ex = 75

Example:Find the value of x to the nearest hundredth.

ex = 75

This transfers to the log statement ln 75 = x

and the calculator will tell you x = 4.32

Evaluate:

Evaluate:

Evaluate:

Evaluate:

Evaluate:

Solve:

Solve:

Solve:

Solve:

Assignment:

Pg. 194C.E. #1 – 9 all

W.E. #2 – 14 evens