Basic Math Examples

$y = \frac{1}{5} \cdot (3^{t})$

Step 1

Rewrite the equation as $\frac{1}{5} \cdot 3^{t} = y$ .

$\frac{1}{5} \cdot 3^{t} = y$

Step 2

Multiply both sides of the equation by $5$ .

$5 (\frac{1}{5} \cdot 3^{t}) = 5 y$

Step 3

Simplify the left side.

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Step 3.1

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

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Step 3.1.1

Combine $\frac{1}{5}$ and $3^{t}$ .

$5 \frac{3^{t}}{5} = 5 y$

Step 3.1.2

Cancel the common factor of $5$ .

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Step 3.1.2.1

Cancel the common factor.

$5 \frac{3^{t}}{5} = 5 y$

Step 3.1.2.2

Rewrite the expression.

$3^{t} = 5 y$

Step 4

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

$\ln (3^{t}) = \ln (5 y)$

Step 5

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

$t \ln (3) = \ln (5 y)$

Step 6

Divide each term in $t \ln (3) = \ln (5 y)$ by $\ln (3)$ and simplify.

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Step 6.1

Divide each term in $t \ln (3) = \ln (5 y)$ by $\ln (3)$ .

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

Step 6.2

Simplify the left side.

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Step 6.2.1

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

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Step 6.2.1.1

Cancel the common factor.

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

Step 6.2.1.2

Divide $t$ by $1$ .

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