Algebra Examples

$5 \cdot 10^{\frac{z}{4}} = 32$

Step 1

Divide each term in $5 \cdot 10^{\frac{z}{4}} = 32$ by $5$ and simplify.

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Step 1.1

Divide each term in $5 \cdot 10^{\frac{z}{4}} = 32$ by $5$ .

$\frac{5 \cdot 10^{\frac{z}{4}}}{5} = \frac{32}{5}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $5$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{5 \cdot 10^{\frac{z}{4}}}{5} = \frac{32}{5}$

Step 1.2.1.2

Divide $10^{\frac{z}{4}}$ by $1$ .

$10^{\frac{z}{4}} = \frac{32}{5}$

Step 2

Take the base $10$ logarithm of both sides of the equation to remove the variable from the exponent.

$\log (10^{\frac{z}{4}}) = \log (\frac{32}{5})$

Step 3

Expand the left side.

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Step 3.1

Expand $\log (10^{\frac{z}{4}})$ by moving $\frac{z}{4}$ outside the logarithm.

$\frac{z}{4} \log (10) = \log (\frac{32}{5})$

Step 3.2

Logarithm base $10$ of $10$ is $1$ .

$\frac{z}{4} \cdot 1 = \log (\frac{32}{5})$

Step 3.3

Multiply $\frac{z}{4}$ by $1$ .

$\frac{z}{4} = \log (\frac{32}{5})$

Step 4

Multiply both sides of the equation by $4$ .

$4 \frac{z}{4} = 4 \log (\frac{32}{5})$

Step 5

Simplify the left side.

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Step 5.1

Cancel the common factor of $4$ .

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Step 5.1.1

Cancel the common factor.

$4 \frac{z}{4} = 4 \log (\frac{32}{5})$

Step 5.1.2

Rewrite the expression.

$z = 4 \log (\frac{32}{5})$

Step 6

The result can be shown in multiple forms.

Exact Form:

$z = 4 \log (\frac{32}{5})$

Decimal Form:

$z = 3.22471989 \dots$