Algebra Examples

$4 \cdot 2^{- \frac{t}{5}} = 288$

Step 1

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

$\ln (4 \cdot 2^{- \frac{t}{5}}) = \ln (288)$

Step 2

Expand the left side.

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Step 2.1

Rewrite $\ln (4 \cdot 2^{- \frac{t}{5}})$ as $\ln (4) + \ln (2^{- \frac{t}{5}})$ .

$\ln (4) + \ln (2^{- \frac{t}{5}}) = \ln (288)$

Step 2.2

Expand $\ln (2^{- \frac{t}{5}})$ by moving $- \frac{t}{5}$ outside the logarithm.

$\ln (4) - \frac{t}{5} \ln (2) = \ln (288)$

Step 2.3

Combine $\ln (2)$ and $\frac{t}{5}$ .

$\ln (4) - \frac{\ln (2) t}{5} = \ln (288)$

Step 3

Simplify the left side.

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Step 3.1

Reorder factors in $\ln (4) - \frac{\ln (2) t}{5}$ .

$\ln (4) - \frac{t \ln (2)}{5} = \ln (288)$

Step 4

Reorder $\ln (4)$ and $- \frac{t \ln (2)}{5}$ .

$- \frac{t \ln (2)}{5} + \ln (4) = \ln (288)$

Step 5

Move all the terms containing a logarithm to the left side of the equation.

$- \frac{t \ln (2)}{5} + \ln (4) - \ln (288) = 0$

Step 6

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$- \frac{t \ln (2)}{5} + \ln (\frac{4}{288}) = 0$

Step 7

Cancel the common factor of $4$ and $288$ .

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Step 7.1

Factor $4$ out of $4$ .

$- \frac{t \ln (2)}{5} + \ln (\frac{4 (1)}{288}) = 0$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $4$ out of $288$ .

$- \frac{t \ln (2)}{5} + \ln (\frac{4 \cdot 1}{4 \cdot 72}) = 0$

Step 7.2.2

Cancel the common factor.

$- \frac{t \ln (2)}{5} + \ln (\frac{4 \cdot 1}{4 \cdot 72}) = 0$

Step 7.2.3

Rewrite the expression.

$- \frac{t \ln (2)}{5} + \ln (\frac{1}{72}) = 0$

Step 8

Subtract $\ln (\frac{1}{72})$ from both sides of the equation.

$- \frac{t \ln (2)}{5} = - \ln (\frac{1}{72})$

Step 9

Multiply both sides of the equation by $- 5$ .

$- 5 (- \frac{t \ln (2)}{5}) = - 5 (- \ln (\frac{1}{72}))$

Step 10

Simplify both sides of the equation.

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Step 10.1

Simplify the left side.

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Step 10.1.1

Simplify $- 5 (- \frac{t \ln (2)}{5})$ .

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Step 10.1.1.1

Cancel the common factor of $5$ .

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Step 10.1.1.1.1

Move the leading negative in $- \frac{t \ln (2)}{5}$ into the numerator.

$- 5 \frac{- t \ln (2)}{5} = - 5 (- \ln (\frac{1}{72}))$

Step 10.1.1.1.2

Factor $5$ out of $- 5$ .

$5 (- 1) \frac{- t \ln (2)}{5} = - 5 (- \ln (\frac{1}{72}))$

Step 10.1.1.1.3

Cancel the common factor.

$5 \cdot - 1 \frac{- t \ln (2)}{5} = - 5 (- \ln (\frac{1}{72}))$

Step 10.1.1.1.4

Rewrite the expression.

$- (- t \ln (2)) = - 5 (- \ln (\frac{1}{72}))$

Step 10.1.1.2

Multiply.

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Step 10.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (t \ln (2)) = - 5 (- \ln (\frac{1}{72}))$

Step 10.1.1.2.2

Multiply $t$ by $1$ .

$t \ln (2) = - 5 (- \ln (\frac{1}{72}))$

Step 10.2

Simplify the right side.

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Step 10.2.1

Multiply $- 1$ by $- 5$ .

$t \ln (2) = 5 \ln (\frac{1}{72})$

Step 11

Divide each term in $t \ln (2) = 5 \ln (\frac{1}{72})$ by $\ln (2)$ and simplify.

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Step 11.1

Divide each term in $t \ln (2) = 5 \ln (\frac{1}{72})$ by $\ln (2)$ .

$\frac{t \ln (2)}{\ln (2)} = \frac{5 \ln (\frac{1}{72})}{\ln (2)}$

Step 11.2

Simplify the left side.

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Step 11.2.1

Cancel the common factor of $\ln (2)$ .

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Step 11.2.1.1

Cancel the common factor.

$\frac{t \ln (2)}{\ln (2)} = \frac{5 \ln (\frac{1}{72})}{\ln (2)}$

Step 11.2.1.2

Divide $t$ by $1$ .

$t = \frac{5 \ln (\frac{1}{72})}{\ln (2)}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$t = \frac{5 \ln (\frac{1}{72})}{\ln (2)}$

Decimal Form:

$t = - 30.84962500 \dots$