1. Logarithmic Functions We know: 2 3 =8 and 2 4 =16 But, for what value of x does 2 x = 10? To solve for an exponent, mathematicians defined logarithms. Since 10 is between 8 and 16, x must be between 3 and 4.

2. Definition of Logarithm y = b x if and only if x = log b y b and y are positive real numbers and b ≠ 1 Base Power So, 2 x = 10, from our little example, can be written as: x = log 2 10 Name the base: Name the power: 2 x x = log 2 10 Exponential Form Logarithmic Form

3. Example 1.Rewrite each equation in exponential form. a.log 3 9 = 2 First, write the Base. 3 Then write the power. y = b x x = log b y if and only if 2 This equals to what’s left over. = 9 b.log 8 1 = 0 8 0 = 1 5 −2

4. Example 2.Evaluate the expression. a.log 4 64 y = b x x = log b y if and only ifWhich piece is missing? When evaluating logs, the solution is the power that makes the log a true statement. log 4 64= ? Rewrite the equation in exponential form. 4 ? = 64log 4 64= 3 b.log 2 0.125 2 ? = −3 c.log 1/4 256 ? = −4 log 1/4 256 d.log 32 232 ? = 2log 32 2

5. Example 2.Evaluate the expression using the calculator. a.log 4 64 MATH A allows you to enter the base and the number to get the answer. MATH A 4 (64) = 3 b.log 2 0.125 c.log 1/4 256 d.log 32 2 MATH A 2 (0.125) = -3 MATH A 1/4 (256) = -4

6. Look at the definition of a logarithm again. y = b x if and only if x = log b y Exponential and Logarithmic functions are INVERSES of each other!!! This means that the domain and range switch places!! Logarithms always have a RANGE of all real numbers and a limited domain. Logarithms have vertical asymptotes. Exponential expressions always have a DOMAIN of all real numbers and a limited range. Exponentials have horizontal asymptotes.

7. ALWAYS All Reals!! Never None NOTE: Your calculator cannot draw the vertical asymptote, so it appears as though the graph stops at x = 2; it does not!! The graph continues down forever; the range is all real numbers. Keep this in mind at all times!!

8. ALWAYS All Reals!! Never None Remember: Exponential functions are INVERSES of logarithms, so the domains & ranges switch. The domain of an exponential function is always all real numbers.

9. This makes the domain all real #’s This makes the range all real #’s