Lesson 5-5 Logarithms

Logarithmic functions

The inverse of the exponential function.

Logarithmic functions The inverse of the exponential function. Basic exponential function: f(x) = b x

Logarithmic functions The inverse of the exponential function. Basic exponential function: f(x) = b x

Logarithmic functions The inverse of the exponential function. Basic logarithmic function: f -1 (x) = log b x

Logarithmic functions The inverse of the exponential function. Basic logarithmic function: f -1 (x) = log b x

Logarithmic functions The inverse of the exponential function. Basic logarithmic function: f -1 (x) = log b x Every (x,y)  (y,x)

Logarithmic functions Basic rule for changing exponential equations to logarithmic equations (or vice-versa):

Logarithmic functions Basic rule for changing exponential equations to logarithmic equations (or vice-versa): log b x = a  b a = x

Logarithmic functions Basic rule for changing exponential equations to logarithmic equations (or vice-versa): log b x = a  b a = x The base of the logarithmic form becomes the base of the exponential form.

Logarithmic functions Basic rule for changing exponential equations to logarithmic equations (or vice-versa): log b x = a  b a = x The answer to the log statement becomes the power in the exponential form.

Logarithmic functions Basic rule for changing exponential equations to logarithmic equations (or vice-versa): log b x = a  b a = x The number you are to take the log of in the log form, becomes the answer in the exponential form.

Examples:

log 5 25 = 2 because 5 2 = 25

Examples: log 5 25 = 2 because 5 2 = 25 log = 3 because 5 3 = 125

Examples: log 5 25 = 2 because 5 2 = 25 log = 3 because 5 3 = 125 log 2 (1/8) = - 3 because 2 -3 = 1/8

base b exponential function f(x) = b x

base b exponential function f(x) = b x Domain: All reals Range: All positive reals

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

base b logarithmic function f -1 (x) = log b (x) Domain: All positive reals Range: All reals

Types of Logarithms

There are two special logarithms that your calculator is programmed for:

Types of Logarithms There are two special logarithms that your calculator is programmed for: log 10 (x)  called the common logarithm

Types of Logarithms There are two special logarithms that your calculator is programmed for: log 10 (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 Logarithms There are two special logarithms that your calculator is programmed for: So, log 10 (x)  log (x) = k if 10 k = x

Types of Logarithms There are two special logarithms that your calculator is programmed for: log e (x)  called the natural logarithm

Types of Logarithms There are two special logarithms that your calculator is programmed for: log e (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 Logarithms There are two special logarithms that your calculator is programmed for: So, log e (x)  ln (x) = k if e k = x

Examples:

log 6.3 = 0.8 because = 6.3

Examples: log 6.3 = 0.8 because = 6.3 ln 5 = 1.6 because e 1.6 = 5

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. 10 x = 75

Example: Find the value of x to the nearest hundredth. 10 x = 75 This transfers to the log statement log = x and the calculator will tell you x = 1.88

Example: Find the value of x to the nearest hundredth. e x = 75

Example: Find the value of x to the nearest hundredth. e x = 75 This transfers to the log statement ln 75 = x and the calculator will tell you x = 4.32

Evaluate:

Solve:

Assignment: Pg. 194 C.E.  #1 – 9 all W.E.  #2 – 14 evens