Trigonometry Examples

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

Step 1

Simplify $\ln (5 e^{t})$ .

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Step 1.1

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

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

Step 1.2

Use logarithm rules to move $t$ out of the exponent.

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

Step 1.3

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

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

Step 1.4

Multiply $t$ by $1$ .

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

Step 2

Move all terms containing $t$ to the left side of the equation.

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Step 2.1

Subtract $5 t$ from both sides of the equation.

$\ln (5) + t - 5 t = 0$

Step 2.2

Subtract $5 t$ from $t$ .

$\ln (5) - 4 t = 0$

Step 3

Subtract $\ln (5)$ from both sides of the equation.

$- 4 t = - \ln (5)$

Step 4

Divide each term in $- 4 t = - \ln (5)$ by $- 4$ and simplify.

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Step 4.1

Divide each term in $- 4 t = - \ln (5)$ by $- 4$ .

$\frac{- 4 t}{- 4} = \frac{- \ln (5)}{- 4}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 4$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 4 t}{- 4} = \frac{- \ln (5)}{- 4}$

Step 4.2.1.2

Divide $t$ by $1$ .

$t = \frac{- \ln (5)}{- 4}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Dividing two negative values results in a positive value.

$t = \frac{\ln (5)}{4}$

Step 4.3.2

Rewrite $\frac{\ln (5)}{4}$ as $\frac{1}{4} \ln (5)$ .

$t = \frac{1}{4} \ln (5)$

Step 4.3.3

Simplify $\frac{1}{4} \ln (5)$ by moving $\frac{1}{4}$ inside the logarithm.

$t = \ln (5^{\frac{1}{4}})$

Step 5

The result can be shown in multiple forms.

Exact Form:

$t = \ln (5^{\frac{1}{4}})$

Decimal Form:

$t = 0.40235947 \dots$