Warm Up Simplify. x 3w z x – 1 1. log10x 2. logbb3w 3. 10log z 4. blogb(x –1) x – 1

Lesson 8.5 Natural Logarithms LO: How to evaluate natural logarithmic expressions and graph natural logarithmic functions.

Exponential functions with e as a base have the same properties as the functions you have studied. The graph of f(x) = ex is like other graphs of exponential functions, such as f(x) = 3x. The domain of f(x) = ex is all real numbers. The range is {y|y > 0}.

The decimal value of e looks like it repeats: 2 The decimal value of e looks like it repeats: 2.718281828… The value is actually 2.71828182890… There is no repeating portion. Caution

Example 1: Graphing Exponential Functions Graph f(x) = ex–2 + 1. Make a table. Because e is irrational, the table values are rounded to the nearest tenth. x –2 –1 1 2 3 4 f(x) = ex–2 + 1 1.0 1.1 1.4 3.7 8.4

Example 2 Graph f(x) = ex – 3. Make a table. Because e is irrational, the table values are rounded to the nearest tenth. x –4 –3 –2 –1 1 2 f(x) = ex – 3 –2.9 –2.7 –0.3 4.4

A logarithm with a base of e is called a natural logarithm and is abbreviated as “ln” (rather than as loge). Natural logarithms have the same properties as log base 10 and logarithms with other bases. The natural logarithmic function f(x) = ln x is the inverse of the natural exponential function f(x) = ex.

The domain of f(x) = ln x is {x|x > 0}. The range of f(x) = ln x is all real numbers. All of the properties of logarithms from Lesson 8-3 also apply to natural logarithms.

Example 3: Simplifying Expression with e or ln A. ln e0.15t B. e3ln(x +1) ln e0.15t = 0.15t e3ln(x +1) = (x + 1)3 C. ln e2x + ln ex ln e2x + ln ex = 2x + x = 3x

Example 4 Simplify. a. ln e3.2 b. e2lnx ln e3.2 = 3.2 e2lnx = x2 c. ln ex +4y ln ex + 4y = x + 4y

Example 5: Using the logarithmic properties to evaluate natural logarithms ln 1 ln 𝑒 2 ln 𝑒 ln 𝑒 −3

Example 5: Expanding and condensing natural logarithms ln 3𝑥 ln 𝑥 3 𝑦 ln 4𝑥 𝑦 2 ln 5𝑡 3 𝑥 2 Condense ln 𝑥 − ln 2 ln 𝑥 −(2 ln 4 + ln 𝑦 ) ln 𝑥 +4 ln 𝑦 − ln 5

Example 6: Sketching the graph of a natural logarithmic function. Sketch the graph of 𝑓 𝑥 . 𝑓 𝑥 =3 − ln (𝑥 −2) 𝑓 𝑥 = ln (𝑥− 3 2 ) 𝑓 𝑥 =− 3 2 ln 𝑥

Example 7: Using the Change of Base formula Use a calculator to evaluate the expression. 𝑙𝑜𝑔 3 12 𝑙𝑜𝑔 4 13 𝑙𝑜𝑔 7 123 𝑙𝑜𝑔 5 18

Homework Pg. 429 # 5 – 35 odd