Solve for t 6*e^(0.25t)=9

$6 \cdot e^{0.25 t} = 9$

Step 1

Divide each term in $(6 \cdot e^{0.25 t}) = 9$ by $6$ and simplify.

Divide each term in $(6 \cdot e^{0.25 t}) = 9$ by $6$ .

$\frac{(6 \cdot e^{0.25 t})}{6} = \frac{9}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 \cdot e^{0.25 t}}{6} = \frac{9}{6}$

Divide $e^{0.25 t}$ by $1$ .

$e^{0.25 t} = \frac{9}{6}$

Simplify the right side.

Cancel the common factor of $9$ and $6$ .

Factor $3$ out of $9$ .

$e^{0.25 t} = \frac{3 (3)}{6}$

Cancel the common factors.

Factor $3$ out of $6$ .

$e^{0.25 t} = \frac{3 \cdot 3}{3 \cdot 2}$

Cancel the common factor.

$e^{0.25 t} = \frac{3 \cdot 3}{3 \cdot 2}$

Rewrite the expression.

$e^{0.25 t} = \frac{3}{2}$

Step 2

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

$\ln (e^{0.25 t}) = \ln (\frac{3}{2})$

Step 3

Expand the left side.

Expand $\ln (e^{0.25 t})$ by moving $0.25 t$ outside the logarithm.

$0.25 t \ln (e) = \ln (\frac{3}{2})$

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

$0.25 t \cdot 1 = \ln (\frac{3}{2})$

Multiply $0.25$ by $1$ .

$0.25 t = \ln (\frac{3}{2})$

Step 4

Divide each term in $0.25 t = \ln (\frac{3}{2})$ by $0.25$ and simplify.

Divide each term in $0.25 t = \ln (\frac{3}{2})$ by $0.25$ .

$\frac{0.25 t}{0.25} = \frac{\ln (\frac{3}{2})}{0.25}$

Simplify the left side.

Cancel the common factor of $0.25$ .

Cancel the common factor.

$\frac{0.25 t}{0.25} = \frac{\ln (\frac{3}{2})}{0.25}$

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{3}{2})}{0.25}$

Simplify the right side.

Replace $e$ with an approximation.

$t = \frac{\log_{2.71828182} (\frac{3}{2})}{0.25}$

Divide $3$ by $2$ .

$t = \frac{\log_{2.71828182} (1.5)}{0.25}$

Log base $2.71828182$ of $1.5$ is approximately $0.4054651$ .

$t = \frac{0.4054651}{0.25}$

Divide $0.4054651$ by $0.25$ .

$t = 1.62186043$

Solve for t 6*e^(0.25t)=9

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