Solve for t 10/(1+4e^(0.01t))=2.5

$\frac{10}{1 + 4 e^{0.01 t}} = 2.5$

Step 1

Multiply both sides by $1 + 4 e^{0.01 t}$ .

$\frac{10}{1 + 4 e^{0.01 t}} (1 + 4 e^{0.01 t}) = 2.5 (1 + 4 e^{0.01 t})$

Step 2

Simplify.

Simplify the left side.

Cancel the common factor of $1 + 4 e^{0.01 t}$ .

Cancel the common factor.

$\frac{10}{1 + 4 e^{0.01 t}} (1 + 4 e^{0.01 t}) = 2.5 (1 + 4 e^{0.01 t})$

Rewrite the expression.

$10 = 2.5 (1 + 4 e^{0.01 t})$

Simplify the right side.

Simplify $2.5 (1 + 4 e^{0.01 t})$ .

Apply the distributive property.

$10 = 2.5 \cdot 1 + 2.5 (4 e^{0.01 t})$

Multiply.

Multiply $2.5$ by $1$ .

$10 = 2.5 + 2.5 (4 e^{0.01 t})$

Multiply $4$ by $2.5$ .

$10 = 2.5 + 10 e^{0.01 t}$

Step 3

Solve for $t$ .

Rewrite the equation as $2.5 + 10 e^{0.01 t} = 10$ .

$2.5 + 10 e^{0.01 t} = 10$

Move all terms not containing $t$ to the right side of the equation.

Subtract $2.5$ from both sides of the equation.

$10 e^{0.01 t} = 10 - 2.5$

Subtract $2.5$ from $10$ .

$10 e^{0.01 t} = 7.5$

Divide each term in $10 e^{0.01 t} = 7.5$ by $10$ and simplify.

Divide each term in $10 e^{0.01 t} = 7.5$ by $10$ .

$\frac{10 e^{0.01 t}}{10} = \frac{7.5}{10}$

Simplify the left side.

Cancel the common factor of $10$ .

Cancel the common factor.

$\frac{10 e^{0.01 t}}{10} = \frac{7.5}{10}$

Divide $e^{0.01 t}$ by $1$ .

$e^{0.01 t} = \frac{7.5}{10}$

Simplify the right side.

Divide $7.5$ by $10$ .

$e^{0.01 t} = 0.75$

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

$\ln (e^{0.01 t}) = \ln (0.75)$

Expand the left side.

Expand $\ln (e^{0.01 t})$ by moving $0.01 t$ outside the logarithm.

$0.01 t \ln (e) = \ln (0.75)$

The natural logarithm of $e$ is $1$ .

$0.01 t \cdot 1 = \ln (0.75)$

Multiply $0.01$ by $1$ .

$0.01 t = \ln (0.75)$

Divide each term in $0.01 t = \ln (0.75)$ by $0.01$ and simplify.

Divide each term in $0.01 t = \ln (0.75)$ by $0.01$ .

$\frac{0.01 t}{0.01} = \frac{\ln (0.75)}{0.01}$

Simplify the left side.

Cancel the common factor of $0.01$ .

Cancel the common factor.

$\frac{0.01 t}{0.01} = \frac{\ln (0.75)}{0.01}$

Divide $t$ by $1$ .

$t = \frac{\ln (0.75)}{0.01}$

Simplify the right side.

Replace $e$ with an approximation.

$t = \frac{\log_{2.71828182} (0.75)}{0.01}$

Log base $2.71828182$ of $0.75$ is approximately $- 0.28768207$ .

$t = \frac{- 0.28768207}{0.01}$

Divide $- 0.28768207$ by $0.01$ .

$t = - 28.76820724$

Solve for t 10/(1+4e^(0.01t))=2.5

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