Algebra Examples

$10 \cdot 3^{\frac{5 t}{4}} = 800$

Step 1

Divide each term in $10 \cdot 3^{\frac{5 t}{4}} = 800$ by $10$ and simplify.

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Step 1.1

Divide each term in $10 \cdot 3^{\frac{5 t}{4}} = 800$ by $10$ .

$\frac{10 \cdot 3^{\frac{5 t}{4}}}{10} = \frac{800}{10}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $10$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{10 \cdot 3^{\frac{5 t}{4}}}{10} = \frac{800}{10}$

Step 1.2.1.2

Divide $3^{\frac{5 t}{4}}$ by $1$ .

$3^{\frac{5 t}{4}} = \frac{800}{10}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $800$ by $10$ .

$3^{\frac{5 t}{4}} = 80$

Step 2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (3^{\frac{5 t}{4}}) = \ln (80)$

Step 3

Expand the left side.

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Step 3.1

Expand $\ln (3^{\frac{5 t}{4}})$ by moving $\frac{5 t}{4}$ outside the logarithm.

$\frac{5 t}{4} \ln (3) = \ln (80)$

Step 3.2

Combine $\frac{5 t}{4}$ and $\ln (3)$ .

$\frac{5 t \ln (3)}{4} = \ln (80)$

Step 4

Multiply both sides of the equation by $\frac{1}{5 \frac{\ln (3)}{4}}$ .

$\frac{1}{5 \frac{\ln (3)}{4}} \cdot \frac{5 t \ln (3)}{4} = \frac{1}{5 \frac{\ln (3)}{4}} \ln (80)$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $\frac{1}{5 \frac{\ln (3)}{4}} \cdot \frac{5 t \ln (3)}{4}$ .

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Step 5.1.1.1

Cancel the common factor of $5$ .

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Step 5.1.1.1.1

Factor $5$ out of $5 \frac{\ln (3)}{4}$ .

$\frac{1}{5 (\frac{\ln (3)}{4})} \cdot \frac{5 t \ln (3)}{4} = \frac{1}{5 \frac{\ln (3)}{4}} \ln (80)$

Step 5.1.1.1.2

Factor $5$ out of $5 t \ln (3)$ .

$\frac{1}{5 (\frac{\ln (3)}{4})} \cdot \frac{5 (t \ln (3))}{4} = \frac{1}{5 \frac{\ln (3)}{4}} \ln (80)$

Step 5.1.1.1.3

Cancel the common factor.

$\frac{1}{5 \frac{\ln (3)}{4}} \cdot \frac{5 (t \ln (3))}{4} = \frac{1}{5 \frac{\ln (3)}{4}} \ln (80)$

Step 5.1.1.1.4

Rewrite the expression.

$\frac{1}{\frac{\ln (3)}{4}} \cdot \frac{t \ln (3)}{4} = \frac{1}{5 \frac{\ln (3)}{4}} \ln (80)$

Step 5.1.1.2

Multiply $\frac{1}{\frac{\ln (3)}{4}}$ by $\frac{t \ln (3)}{4}$ .

$\frac{t \ln (3)}{\frac{\ln (3)}{4} \cdot 4} = \frac{1}{5 \frac{\ln (3)}{4}} \ln (80)$

Step 5.1.1.3

Combine $\frac{\ln (3)}{4}$ and $4$ .

$\frac{t \ln (3)}{\frac{\ln (3) \cdot 4}{4}} = \frac{1}{5 \frac{\ln (3)}{4}} \ln (80)$

Step 5.1.1.4

Cancel the common factor of $4$ .

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Step 5.1.1.4.1

Cancel the common factor.

$\frac{t \ln (3)}{\frac{\ln (3) \cdot 4}{4}} = \frac{1}{5 \frac{\ln (3)}{4}} \ln (80)$

Step 5.1.1.4.2

Divide $\ln (3)$ by $1$ .

$\frac{t \ln (3)}{\ln (3)} = \frac{1}{5 \frac{\ln (3)}{4}} \ln (80)$

Step 5.1.1.5

Cancel the common factor of $\ln (3)$ .

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Step 5.1.1.5.1

Cancel the common factor.

$\frac{t \ln (3)}{\ln (3)} = \frac{1}{5 \frac{\ln (3)}{4}} \ln (80)$

Step 5.1.1.5.2

Divide $t$ by $1$ .

$t = \frac{1}{5 \frac{\ln (3)}{4}} \ln (80)$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{1}{5 \frac{\ln (3)}{4}} \ln (80)$ .

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Step 5.2.1.1

Combine $5$ and $\frac{\ln (3)}{4}$ .

$t = \frac{1}{\frac{5 \ln (3)}{4}} \ln (80)$

Step 5.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$t = 1 \frac{4}{5 \ln (3)} \ln (80)$

Step 5.2.1.3

Multiply $\frac{4}{5 \ln (3)}$ by $1$ .

$t = \frac{4}{5 \ln (3)} \ln (80)$

Step 5.2.1.4

Combine $\frac{4}{5 \ln (3)}$ and $\ln (80)$ .

$t = \frac{4 \ln (80)}{5 \ln (3)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$t = \frac{4 \ln (80)}{5 \ln (3)}$

Decimal Form:

$t = 3.19095402 \dots$