Answer:
a)3
b)13/4
c)-4
d)5
NOTE: You might want to read the first section below the "Step-by-step explanation:" to see if I have interpreted your problems correctly.
Step-by-step explanation:
a)
[tex]\log_x(243)=5[/tex]
b)
[tex]2^{4x-4}=512[/tex]
c)
[tex]5^x=\frac{1}{625}[/tex]
d)
[tex]2^{3x-9}=64[/tex]
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a)
Let's writ ether the logarithmic form in equivalent exponential form:
[tex]x^5=243[/tex]
To solve this we need to take the fifth root of both sides:
[tex]x=243^\frac{1}{5}[/tex]
[tex]x=3[/tex]
b)
We are going to write both sides so their bases are 2.
The left hand side is already base 2 so we are not doing anything to that side.
The 512 however can be written as 2^9.
So we have:
[tex]2^{4x-4}=2^{9}[/tex]
Since the bases are the same, the only thing we can do is set the exponents equal so those are the same as well.
[tex]4x-4=9[/tex]
Add 4 on both sides:
[tex]4x=13[/tex]
Divide both sides by 4:
[tex]x=\frac{13}{4}[/tex]
c)
We are going to write both sides so their bases are 5.
This does not effect left hand side since the base is already 5 on that side.
I know 5^4=625 so 5^(-4)=1/625.
So we have:
[tex]5^x=5^{-4}[/tex]
This implies [tex]x=-4[/tex].
d)
We are going to write both sides so they have base 2.
Left hand side is done. Let's move on to the right. 64=2^6.
[tex]2^{3x-9}=2^6[/tex].
This implies:
[tex]3x-9=6[/tex]
Add 9 on both sides:
[tex]3x=15[/tex]
Divide both sides by 3:
[tex]x=\frac{15}{3}[/tex]
Simplify:
[tex]x=5[/tex]
