4.3 (1) logarithmic functions

Question

The function f(x) = log₁₆(x)

 is the logarithm function with base ____________. So f(16) =  __________, f(1)=_____________, f(1/16)____________, f(25)=_____________, and f(4)___________.

Answer 1

The function f(x) = log₁₆(x)

is the logarithm function with base ____________. So f(16) =  __________, f(1)=_____________, f(1/16)____________, f(25)=_____________, and f(4)___________.

takeaways:

*definition of base: log_bn where b is the base

*process for solving for x when x is the exponent:

c^x=d

xln(c)=ln(d)

x=ln(d)/ln(c)

1) according to the definition above, 16 is the base

2)

f(16)= log₁₆16

To solve this, you need to ask yourself, "16 (the value of the base) to the what power equals 16 (the value of

x)"?

The answer is 16 to the power of 1 is 16. Therefore the answer is 1.

3)

f(1)=log₁₆1

To solve this, you need to ask yourself, "16 (the value of the base) to the what power equals 1 (the value of

x)"?

The answer is 16 to the power of 0 is 1. Therefore the answer is 0.

4)

f(1/16)=log₁₆1/16

To solve this, you need to ask yourself, "16 (the value of the base) to what power equals 1/16"

The answer is 16 to the negative 1 power is 1/16. Therefore the answer is -1.

5)

f(25)=log₁₆25

Plug this into the calculator.

The answer is 1.16

6)

f(4)=log₁₆4

The answer is 1/2.

Answer 2

The function f(x)=log⁡₁₆(x) is the logarithm function with base 16.

1. f(16) = log⁡₁₆(16) = 1 because 16¹ = 16.
2. f(1) = log⁡₁₆(1) = 0 because 16⁰ = 1.
3. f(1/16) = log⁡₁₆(1/16) = -1 because 16^-1 = 1/16.
4. f(25) = log⁡₁₆(25) = 1.161 because 16^1.161 = 25. (You'll need a calculator for this one)
5. 16^x = 25
6. xln(16) = ln(25) [taking the natural log of both sides]
7. x = ln(25) / ln(16) [this is what you would type into your calculator]
8. f(4) = log⁡₁₆(4) = 1/2 because 16^1/2 = 4.

Hope this was helpful.