Evaluate 4*2^(-t/5)=288

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

Step 1

Simplify.

Rewrite $4$ as $2^{2}$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2

Divide each term in $2^{2 - \frac{t}{5}} = 288$ by $4$ and simplify.

Divide each term in $2^{2 - \frac{t}{5}} = 288$ by $4$ .

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

Simplify the left side.

Simplify the numerator.

To write $2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Combine $2$ and $\frac{5}{5}$ .

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

Combine the numerators over the common denominator.

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

Multiply $2$ by $5$ .

$\frac{2^{\frac{10 - t}{5}}}{4} = \frac{288}{4}$

Simplify the right side.

Divide $288$ by $4$ .

$\frac{2^{\frac{10 - t}{5}}}{4} = 72$

Step 3

Multiply both sides by $4$ .

$\frac{2^{\frac{10 - t}{5}}}{4} \cdot 4 = 72 \cdot 4$

Step 4

Simplify.

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{2^{\frac{10 - t}{5}}}{4} \cdot 4 = 72 \cdot 4$

Rewrite the expression.

$2^{\frac{10 - t}{5}} = 72 \cdot 4$

Simplify the right side.

Multiply $72$ by $4$ .

$2^{\frac{10 - t}{5}} = 288$

Step 5

Solve for $t$ .

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

$\ln (2^{\frac{10 - t}{5}}) = \ln (288)$

Expand the left side.

Expand $\ln (2^{\frac{10 - t}{5}})$ by moving $\frac{10 - t}{5}$ outside the logarithm.

$\frac{10 - t}{5} \ln (2) = \ln (288)$

Combine $\frac{10 - t}{5}$ and $\ln (2)$ .

$\frac{(10 - t) \ln (2)}{5} = \ln (288)$

Simplify the left side.

Reorder factors in $\frac{(10 - t) \ln (2)}{5}$ .

$\frac{\ln (2) (10 - t)}{5} = \ln (288)$

Multiply both sides by $5$ .

$\frac{\ln (2) (10 - t)}{5} \cdot 5 = \ln (288) \cdot 5$

Simplify.

Simplify the left side.

Simplify $\frac{\ln (2) (10 - t)}{5} \cdot 5$ .

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{\ln (2) (10 - t)}{5} \cdot 5 = \ln (288) \cdot 5$

Rewrite the expression.

$\ln (2) (10 - t) = \ln (288) \cdot 5$

Apply the distributive property.

$\ln (2) \cdot 10 + \ln (2) (- t) = \ln (288) \cdot 5$

Simplify the expression.

Move $10$ to the left of $\ln (2)$ .

$10 \cdot \ln (2) + \ln (2) (- t) = \ln (288) \cdot 5$

Rewrite using the commutative property of multiplication.

$10 \cdot \ln (2) - \ln (2) t = \ln (288) \cdot 5$

Move $\ln (2)$ .

$10 \ln (2) - t \ln (2) = \ln (288) \cdot 5$

Reorder $10 \ln (2)$ and $- t \ln (2)$ .

$- t \ln (2) + 10 \ln (2) = \ln (288) \cdot 5$

Simplify the right side.

Move $5$ to the left of $\ln (288)$ .

$- t \ln (2) + 10 \ln (2) = 5 \ln (288)$

Solve for $t$ .

Subtract $10 \ln (2)$ from both sides of the equation.

$- t \ln (2) = 5 \ln (288) - 10 \ln (2)$

Divide each term in $- t \ln (2) = 5 \ln (288) - 10 \ln (2)$ by $- 1 \ln (2)$ and simplify.

Divide each term in $- t \ln (2) = 5 \ln (288) - 10 \ln (2)$ by $- 1 \ln (2)$ .

$\frac{- t \ln (2)}{- 1 \ln (2)} = \frac{5 \ln (288)}{- 1 \ln (2)} + \frac{- 10 \ln (2)}{- 1 \ln (2)}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{t \ln (2)}{\ln (2)} = \frac{5 \ln (288)}{- 1 \ln (2)} + \frac{- 10 \ln (2)}{- 1 \ln (2)}$

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

Cancel the common factor.

$\frac{t \ln (2)}{\ln (2)} = \frac{5 \ln (288)}{- 1 \ln (2)} + \frac{- 10 \ln (2)}{- 1 \ln (2)}$

Divide $t$ by $1$ .

$t = \frac{5 \ln (288)}{- 1 \ln (2)} + \frac{- 10 \ln (2)}{- 1 \ln (2)}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$t = - \frac{5 \ln (288)}{\ln (2)} + \frac{- 10 \ln (2)}{- 1 \ln (2)}$

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

Cancel the common factor.

$t = - \frac{5 \ln (288)}{\ln (2)} + \frac{- 10 \ln (2)}{- 1 \ln (2)}$

Rewrite the expression.

$t = - \frac{5 \ln (288)}{\ln (2)} + \frac{- 10}{- 1}$

Move the negative one from the denominator of $\frac{- 10}{- 1}$ .

$t = - \frac{5 \ln (288)}{\ln (2)} - 1 \cdot - 10$

Rewrite $- 1 \cdot - 10$ as $- - 10$ .

$t = - \frac{5 \ln (288)}{\ln (2)} - - 10$

Multiply $- 1$ by $- 10$ .

$t = - \frac{5 \ln (288)}{\ln (2)} + 10$

Step 6

The result can be shown in multiple forms.

Exact Form:

$t = - \frac{5 \ln (288)}{\ln (2)} + 10$

Decimal Form:

$t = - 30.84962500 \dots$

Evaluate 4*2^(-t/5)=288

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