Exponential and Logarithmic Functions Chapter 5 Exponential and Logarithmic Functions

Section 4 Logarithmic Functions

Logarithmic Functions are inverses of exponents Where logs are in the form y = logax (log base a of x equals y) Written as an exponent ay = x

Examples of relating logs to exponents a) y = log3x  3y = x b) 4 = log381  34 = 81 c) -1 = log5(1/5)  5-1 = 1/5 *the base of the log = base of exponent, a, never changes

Converting Exponents into Logs a) 1 Converting Exponents into Logs a) 1.23 = m b) eb = 9 c) a4 = 24 Converting Logs into Exponents Loga4 = 5 -3 = Logeb Log35 = c

Finding Exact Values of Logs log216 log3(1/27) log10√10

Keeping in mind that logs are inverses of exponents: Exponent functions therefore logarithmic functions Domain: all real Domain: {x| x > 0} Range: {y| y > 0} Range: all real

Find domain of logs: Take “of” part and set > 0 and solve for x log2(x + 3) log2|x| 3 – 2log4[(x/2) – 5]

Graphing Logs: Exponential basic points: (-1, 1/a) (0, 1) (1, a) ; H.A.: y = 0 then Logarithmic basic points: (1/a, -1) (1, 0) (a, 1); V.A.: x = 0 *properties will be “reversed”

Exponent properties Logarithmic properties D: all real D: x > 0 R: y > 0 R: all real H.A.: y = 0 V.A.: x = 0 y-int: (0, 1) x-int: (1, 0) a > 1 0 < a < 1 a > 1 0 < a < 1

Example: Graphing Using Transformations f(x) = 3 + log2(x + 2)

Example 8: Solving a Logarithmic Equation (bases need to be positive – discard extraneous answers if necessary) Solve: log3(4x – 7) = 2 logx64 = 2

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