Exponential and logarithmic function, Lecture notes of Mathematics

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IMAT Math

1. EXPONENTIAL FUNCTION DAVID. PARK

Exponential rule

If 𝑛 is a positive integer, than 𝑎 𝑛 is the product of n factors of 𝑎. We say that a is the base and 𝑛 is the exponent 𝑎 𝑛 = 𝑎 × 𝑎 × 𝑎 ⋯ × 𝑎

Exponential rule

Simplify (1) ( 3 4

5 ÷ 81 2 × ( 2 3

3 (2) ( 2 7

2 × 16 3 × ( 7 2

2

Exponential rule (Radical)

If 𝑎 > 0 and 𝑚, 𝑛 are integer larger than 2, than 𝑎 1 𝑛 (^) = 𝑛 𝑎 Is called the n-th root of 𝑎 And 𝑎 𝑚 𝑛 (^) = 𝑛 𝑎 𝑚 Is called the n-th root of a power of m

Exponential rule

Simplify (1) 3 𝑎 2 𝑏 4 × 4 𝑎 3 𝑏 2 × 𝑎 4 𝑏 3 (2) 3 16 × (^4 ) 16 + ( 3 25 ) 2 3 5

Exponential rule

If 𝑥 𝑥 = 𝑥 𝑎 , then what is the value of 𝑎?

Exponential rule

If an expression has a radical (usually square root), its conjugate is formed by changing the sign between terms. If 𝑥 = 𝑎 + 𝑏, then its conjugate is 𝑥 = 𝑎 − 𝑏

Exponential rule

General rationalizing denominator 𝑐 𝑎+ 𝑏

𝑐×( 𝑎+ 𝑏 ) ( 𝑎+ 𝑏)×( 𝑎+ 𝑏 )

𝑐×(𝑎+ 𝑏 ) 𝑎−𝑏

Exponential function

1. Domain: All real number
2. Range: All positive real number
3. one-to-one function

Exponential function

1. When 𝑎 > 1 , as 𝑥 increases, 𝑦 also increases
2. When 0 < 𝑎 < 1 , as 𝑥 increases, 𝑦 decreases
3. The graph passes through the point 0 , 1
4. Horizontal asymptote is 𝑦 = 0

Exponential function

The function 𝑓 𝑥 is defined by 𝑓 𝑥 = −𝑎 𝑥

• 𝑏 where 𝑎 and 𝑏 are constants. In the xy plane, the graph of y = 𝑓 𝑥 − 12 has a y-intercept at ( 0 , − 75 7 ). The product of 𝑎 and 𝑏 is 320 7

. What is the value of 𝑎?

‘𝑒‘ is an important Mathematical constant and an irrational number. The value of ‘𝑒‘ is approximately 2.

Exponential function

Exponential equation

An exponential equation is an equation in which the variable appears in the exponent If 𝑎 > 0 , 𝑎 ≠ 1 , 𝑎 𝑓(𝑥) = 𝑎 𝑔(𝑥) , then 𝑓 𝑥 = 𝑔(𝑥)

Exponential equation

Solve the equations (1) 2 2𝑥 × 2 𝑥+ 1 = 16 (2) 3 𝑥+ 1

• 3 𝑥 = 36