Algebra Examples

$y = a {(0.5)}^{\frac{t}{h}}$

Step 1

Rewrite the equation as $a \cdot {0.5}^{\frac{t}{h}} = y$ .

$a \cdot {0.5}^{\frac{t}{h}} = y$

Step 2

Divide each term in $a \cdot {0.5}^{\frac{t}{h}} = y$ by $a$ and simplify.

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Step 2.1

Divide each term in $a \cdot {0.5}^{\frac{t}{h}} = y$ by $a$ .

$\frac{a \cdot {0.5}^{\frac{t}{h}}}{a} = \frac{y}{a}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $a$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{a \cdot {0.5}^{\frac{t}{h}}}{a} = \frac{y}{a}$

Step 2.2.1.2

Divide ${0.5}^{\frac{t}{h}}$ by $1$ .

${0.5}^{\frac{t}{h}} = \frac{y}{a}$

Step 3

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

$\ln ({0.5}^{\frac{t}{h}}) = \ln (\frac{y}{a})$

Step 4

Expand the left side.

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Step 4.1

Expand $\ln ({0.5}^{\frac{t}{h}})$ by moving $\frac{t}{h}$ outside the logarithm.

$\frac{t}{h} \ln (0.5) = \ln (\frac{y}{a})$

Step 4.2

Combine $\frac{t}{h}$ and $\ln (0.5)$ .

$\frac{t \ln (0.5)}{h} = \ln (\frac{y}{a})$

Step 5

Multiply both sides by $h$ .

$\frac{t \ln (0.5)}{h} h = \ln (\frac{y}{a}) h$

Step 6

Simplify the left side.

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Step 6.1

Cancel the common factor of $h$ .

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Step 6.1.1

Cancel the common factor.

$\frac{t \ln (0.5)}{h} h = \ln (\frac{y}{a}) h$

Step 6.1.2

Rewrite the expression.

$t \ln (0.5) = \ln (\frac{y}{a}) h$

Step 7

Divide each term in $t \ln (0.5) = \ln (\frac{y}{a}) h$ by $\ln (0.5)$ and simplify.

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Step 7.1

Divide each term in $t \ln (0.5) = \ln (\frac{y}{a}) h$ by $\ln (0.5)$ .

$\frac{t \ln (0.5)}{\ln (0.5)} = \frac{\ln (\frac{y}{a}) h}{\ln (0.5)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $\ln (0.5)$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{t \ln (0.5)}{\ln (0.5)} = \frac{\ln (\frac{y}{a}) h}{\ln (0.5)}$

Step 7.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{y}{a}) h}{\ln (0.5)}$