One-to-One Property

[nycmathguy]

Section 3.1

The One-to-One Property states

For a > 0 and a ≠ 1, a^x = a^y
if and only if x = y.

Use the One-to-One Property to solve the equation for x.

a. 8 = 2^(2x−1)

b. (1/3)^(-x) = 27

Part (a)

I have to rewrite 8 having a base 2.

Let 8 = 2^3

I can now apply the One-to-One Property.

2^3 = 2^(2x - 1)

3 = 2x - 1

3 + 1 = 2x

4 = 2x

4/2 = x

2 = x

Yes?

Part (b)

(1/3)^(-x) = 3^x.

3^x = 27

Let 27 = 3^3.

3^x = 3^3

Bases are the same. Bring down exponents.

x = 3

Yes?

 

[MathLover1]

correct

 

[nycmathguy]

correct

Are exponential and logarithmic functions opposite operations?

 

[MathLover1]

Logarithmic functions are the inverses of exponential functions. The inverse of the exponential function y = a^x is x = a^y. The logarithmic function y = log(a, x ) is defined to be equivalent to the exponential equation x = a^y.

 

[nycmathguy]

Logarithmic functions are the inverses of exponential functions. The inverse of the exponential function y = a^x is x = a^y. The logarithmic function y = log(a, x ) is defined to be equivalent to the exponential equation x = a^y.

Thank you. I think logarithmic functions is the next section.