Logarithmic Functions & Graphs, Lesson 3.2, page 388 Objective: To graph logarithmic functions, to convert between exponential and logarithmic equations, and find common and natural logarithms using a calculator.

DEFINITION Logarithmic function – inverse of exponential function If y = b x, then the inverse is x = b y So y is the power which we raise b to in order to get x. Since we can’t solve this for y, we change it to logarithmic form which is y = log b x

Think of logs like this… log b N = P and b p = N Key: b = base, N = number, P = power Restrictions: b > 0 and b cannot equal 1 *N > 0 because the log of zero or a negative number is undefined.

Changing Exponential  Log Log form => log b N = P Ex) log 2 8 = 3 Think: A logarithm equals an exponent! Exponential form => b P = N Ex) 2 3 = 8

Examples of Conversion Log Form: log b N = P Exponential Form: b P = N Log 2 64 = 6 Log = 3 Log 4 16 = = = = 256

Rewrite the following exponential expression as a logarithmic one.

See Example 1, page 389. Check Point 1. Write each equation in its equivalent exponential form: A) 3 = log 7 x B) 2 = log b 25 C) log 4 26 = y

See Example 2, page 389. Check Point 2. Write each equation in its equivalent logarithmic form: A) 2 5 = x B) b 3 = 27 C) e y = 33

See Example 3, page 389. Check Point 3. Evaluate: A) log B) log 3 3 C) log 36 6

See page 390. BASIC LOG PROPERTIES log b b = 1 log b 1 = 0 INVERSE PROPERTIES OF LOGS log b b x = x b log b x = x

Examples Check Point 4. A) log 9 9b) log 8 1 Check Point 5: A) log b) 3 log 3 17

Graphs Since exponential and logarithmic functions are inverses of each other, their graphs are also inverses.

Logarithmic function and exponential function are inverses of each other. The domain of the exponential function is all reals, so that’s the domain of the logarithmic function. The range of the exponential function is x>0, so the range of the logarithmic function is y>0.

See Example 6, page 391. Check Point 6: Graph f(x) = 3 x and g(x) = log 3 x in the same rectangular coordinate system.

Graph f(x) = 3 x. (  3,1/27) 1/27 33 (  2, 1/9) 1/9 22 (  1, 1/3) 1/3 11 (3, 27) y = f(x) = 3 x (2, 9)2 (1, 3)1 (0, 1)0 (x, y)x

Now let’s add f(x) = log 3 x. (Simply find the inverse of each point from f(x)= 3 x.) f(x)= 3 x (0, 1) (1, 3) (2, 9) (3, 27) (  1, 1/3) (  2, 1/9) (  3,1/27)

See Characteristics of Graphs of Logs on page 392. See Table 3.4 on Transformations.

Graphing Summary Logarithmic functions are inverses of exponential functions. Easier if rewrite as an exponential before graphing. 1.Choose values for y. 2.Compute values for x. 3.Plot the points and connect them with a smooth curve. * Note that the curve does not touch or cross the y-axis.

Comparing Exponential and Logarithmic Functions

Domain Restrictions for Logarithmic Functions Since a positive number raised to an exponent (pos. or neg.) always results in a positive value, you can ONLY take the logarithm of a POSITIVE NUMBER. Remember, the question is: What POWER can I raise the base to, to get this value? DOMAIN RESTRICTION:

See Example 7, page 393. Check Point 7: Find the domain of f(x)=log 4 (x-5).

Common Logarithms -- Intro If no value is stated for the base, it is assumed to be base 10. log(1000) means, “What power do I raise 10 to, to get 1000?” The answer is 3. log(1/10) means, “What power do I raise 10 to, to get 1/10?” The answer is -1.

COMMON LOGARITHMS A common logarithm is a log that uses 10 as its base. Log 10 y is written simply as log y. Examples of common logs are Log 100, log 50, log 26.2, log (1/4) Log button on your calculator is the common log *

Find each of the following common logarithms on a calculator. Round to four decimal places. a) log 723,456 b) log c) log (  4)

Find each of the following common logarithms on a calculator. Does not exist ERR: non real ans log (  4)   log log 723,456 RoundedReadoutFunction Value

Natural Logarithms -- Intro ln(x) represents the natural log of x, which has a base=e What is e? If you plug large values into you get closer and closer to e. logarithmic functions that involve base e are found throughout nature Calculators have a button “ln” which represents the natural log.

Natural Logarithms Logarithms, base e, are called natural logarithms. The abbreviation “ln” is generally used for natural logarithms. Thus, ln x means log e x. * ln button on your calculator is the natural log *

Find each of the following natural logarithms on a calculator. Round to four decimal places. a) ln 723,456 b) ln c) ln (  4)

Does not exist ERR: nonreal answer ln (  4)   ln ln 723,456 RoundedReadoutFunction Value