Algebra Examples

$3.9 = \frac{2}{3} \log (\frac{E}{E_{0}})$

Step 1

Rewrite the equation as $\frac{2}{3} \cdot \log (\frac{E}{𝐄_{0}}) = 3.9$ .

$\frac{2}{3} \cdot \log (\frac{E}{𝐄_{0}}) = 3.9$

Step 2

Multiply both sides of the equation by $\frac{3}{2}$ .

$\frac{3}{2} (\frac{2}{3} \cdot \log (\frac{E}{𝐄_{0}})) = \frac{3}{2} \cdot 3.9$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{3}{2} (\frac{2}{3} \cdot \log (\frac{E}{𝐄_{0}}))$ .

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Step 3.1.1.1

Combine $\frac{2}{3}$ and $\log (\frac{E}{𝐄_{0}})$ .

$\frac{3}{2} \cdot \frac{2 \log (\frac{E}{𝐄_{0}})}{3} = \frac{3}{2} \cdot 3.9$

Step 3.1.1.2

Cancel the common factor of $3$ .

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Step 3.1.1.2.1

Cancel the common factor.

$\frac{3}{2} \cdot \frac{2 \log (\frac{E}{𝐄_{0}})}{3} = \frac{3}{2} \cdot 3.9$

Step 3.1.1.2.2

Rewrite the expression.

$\frac{1}{2} (2 \log (\frac{E}{𝐄_{0}})) = \frac{3}{2} \cdot 3.9$

Step 3.1.1.3

Cancel the common factor of $2$ .

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Step 3.1.1.3.1

Factor $2$ out of $2 \log (\frac{E}{𝐄_{0}})$ .

$\frac{1}{2} (2 \log (\frac{E}{𝐄_{0}})) = \frac{3}{2} \cdot 3.9$

Step 3.1.1.3.2

Cancel the common factor.

$\frac{1}{2} (2 \log (\frac{E}{𝐄_{0}})) = \frac{3}{2} \cdot 3.9$

Step 3.1.1.3.3

Rewrite the expression.

$\log (\frac{E}{𝐄_{0}}) = \frac{3}{2} \cdot 3.9$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\frac{3}{2} \cdot 3.9$ .

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Step 3.2.1.1

Multiply $\frac{3}{2} \cdot 3.9$ .

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Step 3.2.1.1.1

Combine $\frac{3}{2}$ and $3.9$ .

$\log (\frac{E}{𝐄_{0}}) = \frac{3 \cdot 3.9}{2}$

Step 3.2.1.1.2

Multiply $3$ by $3.9$ .

$\log (\frac{E}{𝐄_{0}}) = \frac{11.7}{2}$

Step 3.2.1.2

Divide $11.7$ by $2$ .

$\log (\frac{E}{𝐄_{0}}) = 5.85$

Step 4

Rewrite $\log (\frac{E}{𝐄_{0}}) = 5.85$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$10^{5.85} = \frac{E}{𝐄_{0}}$

Step 5

Solve for $E$ .

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Step 5.1

Rewrite the equation as $\frac{E}{𝐄_{0}} = 10^{5.85}$ .

$\frac{E}{𝐄_{0}} = 10^{5.85}$

Step 5.2

Multiply both sides by $𝐄_{0}$ .

$\frac{E}{𝐄_{0}} 𝐄_{0} = 10^{5.85} 𝐄_{0}$

Step 5.3

Simplify.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Cancel the common factor of $𝐄_{0}$ .

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Step 5.3.1.1.1

Cancel the common factor.

$\frac{E}{𝐄_{0}} 𝐄_{0} = 10^{5.85} 𝐄_{0}$

Step 5.3.1.1.2

Rewrite the expression.

$E = 10^{5.85} 𝐄_{0}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Raise $10$ to the power of $5.85$ .

$E = 707945.78438413 𝐄_{0}$