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**Im helping a friend go through this problem, so "the answer will be shown eventually" because im 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 lets start

Ok, lets 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

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 lets start

**APPROACH #1**Ok, lets 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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