- #1
matadorqk
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**Im helping a friend go through this problem, so "the answer will be shown eventually" because I am editing as we go through.
[tex]2^{x-1}=25[/tex]
All Log/Exponent Formulas, they'll be shown as we go.
There are various ways of approaching this problem, so let's start
APPROACH #1
Ok, let's multiply both sides by [tex]\log_{10}[/tex]
So, [tex]\log 2^{x-1}=log 25 [/tex]
We know that [tex] \log a^{b}=b \log a [/tex], so apply this to our left hand side.
So, if we know the above formula, the [tex]\log 2^{x-1}=log a ^{b}[/tex]. So, we find our a and b.
A=2 and B=x-1
So, [tex]log(2)^{x-1}=(x-1)log2[/tex]
Therefore, [tex] (x-1) log 2 = log 25 [/tex] so multiply x-1.
Therefore, [tex] x log 2 - 1 log 2 = log 25 [/tex]
So we solve for x!
So, we pass log 2 to the other side, by adding on each side.
[tex] x log 2 = log 25 + log 2. [/tex]
Divide both sides by log 2.
[tex] x= \frac{log 25 + log 2}{log 2} [/tex]
Now use the calculator.
x=5.64
APPROACH #2
If we know that [tex](a^{b})(a^{c}) = a^{b+c}[/tex]
we know that [tex]2^{x-1}=(2^{x})(2^{-1}).[/tex]
Therefore, [tex](2^{x})(\frac{1}{2})=25[/tex].
So multiply both sides by 2, to cancel out the 1/2.
[tex] 2^{x}=50 [/tex]
Now its simple log work, we multiply both sides by [tex]log_{10}[/tex]
So [tex] log 2^{x} = log 50 [/tex]
So [tex] x log 2 = log 50 [/tex]
So [tex] x=\frac{log 50}{log 2} = 5.64 [/tex]
Homework Statement
[tex]2^{x-1}=25[/tex]
Homework Equations
All Log/Exponent Formulas, they'll be shown as we go.
The Attempt at a Solution
There are various ways of approaching this problem, so let's start
APPROACH #1
Ok, let's multiply both sides by [tex]\log_{10}[/tex]
So, [tex]\log 2^{x-1}=log 25 [/tex]
We know that [tex] \log a^{b}=b \log a [/tex], so apply this to our left hand side.
So, if we know the above formula, the [tex]\log 2^{x-1}=log a ^{b}[/tex]. So, we find our a and b.
A=2 and B=x-1
So, [tex]log(2)^{x-1}=(x-1)log2[/tex]
Therefore, [tex] (x-1) log 2 = log 25 [/tex] so multiply x-1.
Therefore, [tex] x log 2 - 1 log 2 = log 25 [/tex]
So we solve for x!
So, we pass log 2 to the other side, by adding on each side.
[tex] x log 2 = log 25 + log 2. [/tex]
Divide both sides by log 2.
[tex] x= \frac{log 25 + log 2}{log 2} [/tex]
Now use the calculator.
x=5.64
APPROACH #2
If we know that [tex](a^{b})(a^{c}) = a^{b+c}[/tex]
we know that [tex]2^{x-1}=(2^{x})(2^{-1}).[/tex]
Therefore, [tex](2^{x})(\frac{1}{2})=25[/tex].
So multiply both sides by 2, to cancel out the 1/2.
[tex] 2^{x}=50 [/tex]
Now its simple log work, we multiply both sides by [tex]log_{10}[/tex]
So [tex] log 2^{x} = log 50 [/tex]
So [tex] x log 2 = log 50 [/tex]
So [tex] x=\frac{log 50}{log 2} = 5.64 [/tex]
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