CHAPTER logarithmic functions

objectives Write equivalent forms for exponential and logarithmic functions. Write, evaluate, and graph logarithmic functions.

Logarithmic functions ◦ How many times would you have to double $1 before you had $8? You could use an exponential equation to model this situation. 1(2 x ) = 8. You may be able to solve this equation by using mental math if you know 2 3 = 8. So you would have to double the dollar 3 times to have $8. ◦ How many times would you have to double $1 before you had $512? You could solve this problem if you could solve 2x = 8 by using an inverse operation that undoes raising a base to an exponent equation to model this situation. This operation is called finding the logarithm. A logarithm is the exponent to which a specified base is raised to obtain a given value.

Logarithmic functions You can write an exponential equation as a logarithmic equation and vice versa.

Example #1 ◦ Write each exponential equation in logarithmic form. Exponential Equation Logarithmic Form 3 5 = = = 10,000 6 –1 = a b = c

Example #2 ◦ Write each exponential equation in logarithmic form. Exponential Equation Logarithmic Form 9 2 = = 27 x 0 = 1(x ≠ 0)

Student guided practice ◦ Do problems 2 to 5 in your book page 253

Example #3 ◦ Write each logarithmic form in exponential equation. Logarithmic Form Exponential Equation log 9 9 = 1 log = 9 log 8 2 = log 4 = –2 log b 1 = 0

Student guided practice ◦ Do problems 6 to 9 in your book page253

Logarithmic functions ◦ A logarithm with base 10 is called a common logarithm. If no base is written for a logarithm, the base is assumed to be 10. For example, log 5 = log 10 5.

Example #4 ◦ Evaluate by using mental math. log 0.01

Example #5 ◦ Evaluate by using mental math. log 5 125

Example #6 ◦ Evaluate by using mental math. log 5 1 5

Student guided practice ◦ Do problems 10 to13 in your book page 253

Properties of logarithms ◦ 1. log a (uv) = log a u + log a v ◦ 2. log a (u / v) = log a u - log a v ◦ 3. log a u n = n log a u ◦ Many logarithmic expressions may be rewritten, either expanded or condensed, using the three properties above. Expanding is breaking down a complicated expression into simpler components. Condensing is the reverse of this process.

Properties of logs ◦ In less formal terms, the log rules might be expressed as: ◦ 1) Multiplication inside the log can be turned into addition outside the log, and vice versa. ◦ 2) Division inside the log can be turned into subtraction outside the log, and vice versa. ◦ 3) An exponent on everything inside a log can be moved out front as a multiplier, and vice versa.

Example ◦ Expand log 3 (2 x ). Expand log 4 ( 16 / x ).

Student guided practice ◦ Do problems 1-6 in the worksheet

Condensing logs ◦ Simplify log 2 ( x ) + log 2 ( y ). ◦ Simplify log 3 (4) – log 3 (5). ◦ Simplify 3 log 2 ( x ) – 4 log 2 ( x + 3) + log 2 ( y ).

Student guided practice ◦ Do odd problems from in the worksheet

Graphs of log functions ◦ Because logarithms are the inverses of exponents, the inverse of an exponential function, such as y = 2x, is a logarithmic function, such as y = log 2 x. The domain of y = 2x is all real numbers ( R ), and the range is {y|y > 0}. The domain of y = log 2 x is {x|x > 0}, and the range is all real numbers ( R ).

Example #7 ◦ Use the x-values {–2, –1, 0, 1, 2}. Graph the function and its inverse. Describe the domain and range of the inverse function. 1 f(x) = 1.25 x 210–1–2x

Continue ◦ Inverse 210–1–2 f –1 (x) = log 1.25 x x

Example #8 ◦ Use the x-values {–2, –1, 0, 1, 2}. Graph the function and its inverse. Describe the domain and range of the inverse function. f(x) = x 1 2 x–2–1012 f(x) =( ) x 421

Continue ◦ Inverse

Student guided practice ◦ Do problems 14 and 15 in your book page 253

Homework ◦ Do problems in your book page 253

Closure ◦ Today we learned about logarithmic functions ◦ Next class we are going to have. A quiz and review

Have a great day