Solving the exponential equations

Solving the exponential equations

Postby Guest » Mon Aug 23, 2021 9:15 pm

a. [tex](0,3)^{3x - 2} = 1[/tex]

b. [tex](\frac{1}{5})^x = 25[/tex]

c. [tex]2^{x^2 - 3x + 2} = 4[/tex]

d. [tex](0,5)^{x + 7}.(0,5)^{1 - 2x} = 2[/tex]
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Re: Solving the exponential equations

Postby Guest » Sat Feb 26, 2022 8:08 pm

Did you try at all? Some basic knowledge about exponents is really all you need.

For example, you should know that, for any positive number, a, [tex]a^x= 1[/tex] if and only if x= 0.
a) [tex](0,3)^{3x- 2}= 1[/tex] if and only if 3x- 2= 0. So what is x?

you should know that [tex]a^x= a^y[/tex] if and only if x= y.
b) [tex]25= 5^2= \left(\frac{1}{5}\right)^{-2}[/tex] so [tex]\left(\frac{1}{5}\right)^x= \left(\frac{1}{5}\right)^{-2}[/tex]. What is x?

b) Similarly, [tex]2^{x^2- 3x+ 2}= 4= 2^2[/tex] so [tex]x^2- 3x+ 2= 2[/tex]. [tex]x^2- 3x= x(x- 3)= 0[/tex].
What is x?

Okay, d is a little harder. For d you also need to know that [tex](a^x)(a^y)= a^{x+ y}[/tex].
d) [tex](0,5)^{x+ 7}(0,5)^{1- 2x}= (0,5)^{8- x}= 2= \left(\frac{1}{2}\right)^{-1}= (0,5)^{-1}[/tex]
so 8- x= -1. What is x?
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