Calculus Examples

$6 e^{5 t} = 3 e^{2 t}$

Step 1

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

$\ln (6 e^{5 t}) = \ln (3 e^{2 t})$

Step 2

Expand the left side.

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Step 2.1

Rewrite $\ln (6 e^{5 t})$ as $\ln (6) + \ln (e^{5 t})$ .

$\ln (6) + \ln (e^{5 t}) = \ln (3 e^{2 t})$

Step 2.2

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

$\ln (6) + 5 t \ln (e) = \ln (3 e^{2 t})$

Step 2.3

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

$\ln (6) + 5 t \cdot 1 = \ln (3 e^{2 t})$

Step 2.4

Multiply $5$ by $1$ .

$\ln (6) + 5 t = \ln (3 e^{2 t})$

Step 3

Expand the right side.

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Step 3.1

Rewrite $\ln (3 e^{2 t})$ as $\ln (3) + \ln (e^{2 t})$ .

$\ln (6) + 5 t = \ln (3) + \ln (e^{2 t})$

Step 3.2

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

$\ln (6) + 5 t = \ln (3) + 2 t \ln (e)$

Step 3.3

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

$\ln (6) + 5 t = \ln (3) + 2 t \cdot 1$

Step 3.4

Multiply $2$ by $1$ .

$\ln (6) + 5 t = \ln (3) + 2 t$

Step 4

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

$\ln (6) - \ln (3) = - 5 t + 2 t$

Step 5

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

$\ln (\frac{6}{3}) = - 5 t + 2 t$

Step 6

Divide $6$ by $3$ .

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

Step 7

Add $- 5 t$ and $2 t$ .

$\ln (2) = - 3 t$

Step 8

Since $t$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- 3 t = \ln (2)$

Step 9

Divide each term in $- 3 t = \ln (2)$ by $- 3$ and simplify.

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Step 9.1

Divide each term in $- 3 t = \ln (2)$ by $- 3$ .

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

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $- 3$ .

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Step 9.2.1.1

Cancel the common factor.

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

Step 9.2.1.2

Divide $t$ by $1$ .

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

Step 9.3

Simplify the right side.

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Step 9.3.1

Move the negative in front of the fraction.

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

Step 10

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$t = - 0.23104906 \dots$