Warm-Up: Graph the logarithmic function and exponential function then state the domain and range for each. D: x-int: R: y-int: HA: D: x-int: R: y-int: VA: HW pages325-327 (EVEN 12-32, 42-56, 70-88)

3.3 Properties of Logarithms Objective: Change bases of logarithms Apply properties of logarithms Rewrite logarithmic expressions by expansion and contraction

Recall: Properties of Logarithms logaa = 1 logaax = x If loga x = loga y then x = y

Remember Common Logarithm?? Is the “common” logarithm This is the LOG button on the calculator A log is implied to be base ten when we don’t write the base… log107 = log 7

Remember Natural Logarithm?? loge Is the “natural” logarithm This is the ln button on the calculator ln is used to represent loge … loge7 = ln 7

Product Property Express as a sum of logarithms. 1) log43N = log43 + log4N 2) ln6 = ln2 + ln3 3) ln19 + ln3 = ln(193) = ln (57)

Ex. Express as a sum of logarithms, then simplify 4) log2 (416) = log2 4 + log216 = log2 22 + log224 = 2 + 4 = 6

Quotient Property

Ex. Express as the difference of logs 5) 6) 7)

Power Property

Ex. Express as a product. = -5 logb9 8) 9)

Rewrite the logarithm of a quotient 10)

Change of base Formula Let a, b, and x be positive real numbers such that and b do not = 1. then log base a of x can be converted to a different base as follows.

Example Rewrite the logarithm as a ratio of (a) common logs and (b) natural logs.

Example Rewrite as a common log using change of base

Expand. log105x3y log105 + log10x3 + log10y log105 + 3 log10x + log10y

Expand Simplify the division. Simplify the multiplication of 4 Change the radical sign to an exponent Express the exponent as a product

Ex. Condense.

Condense Express all products as exponents Change the fractional exponent to a radical sign. Simplify the subtraction. Simplify the addition.

Summary: Properties of Logarithms because a0 = 1 logaa = 1 because a1 = a logaax = x If loga x = loga y then x = y Product Property Quotient Property Power Property Change-of-Base

Sneedlegrit: Expand: Condense: HW pages325-327 (EVEN 12-32, 42-56, 70-88)