Solve for t 1000(e)^(0.09t)=5000

$1000 {(e)}^{0.09 t} = 5000$

Step 1

Divide each term in $1000 e^{0.09 t} = 5000$ by $1000$ and simplify.

Divide each term in $1000 e^{0.09 t} = 5000$ by $1000$ .

$\frac{1000 e^{0.09 t}}{1000} = \frac{5000}{1000}$

Simplify the left side.

Cancel the common factor of $1000$ .

Cancel the common factor.

$\frac{1000 e^{0.09 t}}{1000} = \frac{5000}{1000}$

Divide $e^{0.09 t}$ by $1$ .

$e^{0.09 t} = \frac{5000}{1000}$

Simplify the right side.

Divide $5000$ by $1000$ .

$e^{0.09 t} = 5$

Step 2

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

$\ln (e^{0.09 t}) = \ln (5)$

Step 3

Expand the left side.

Expand $\ln (e^{0.09 t})$ by moving $0.09 t$ outside the logarithm.

$0.09 t \ln (e) = \ln (5)$

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

$0.09 t \cdot 1 = \ln (5)$

Multiply $0.09$ by $1$ .

$0.09 t = \ln (5)$

Step 4

Divide each term in $0.09 t = \ln (5)$ by $0.09$ and simplify.

Divide each term in $0.09 t = \ln (5)$ by $0.09$ .

$\frac{0.09 t}{0.09} = \frac{\ln (5)}{0.09}$

Simplify the left side.

Cancel the common factor of $0.09$ .

Cancel the common factor.

$\frac{0.09 t}{0.09} = \frac{\ln (5)}{0.09}$

Divide $t$ by $1$ .

$t = \frac{\ln (5)}{0.09}$

Simplify the right side.

Replace $e$ with an approximation.

$t = \frac{\log_{2.71828182} (5)}{0.09}$

Log base $2.71828182$ of $5$ is approximately $1.60943791$ .

$t = \frac{1.60943791}{0.09}$

Divide $1.60943791$ by $0.09$ .

$t = 17.88264347$