Precalculus Examples

$- \frac{\ln (1 + 9 e^{- 0.1656 t})}{\ln (p)} + \frac{3 (\ln (5))}{\ln (p)} + \frac{3 \ln (2)}{\ln (p)}$

Step 1

Combine the numerators over the common denominator.

$\frac{- \ln (1 + 9 e^{- 0.1656 t}) + 3 (\ln (5)) + 3 \ln (2)}{\ln (p)}$

Step 2

Simplify each term.

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Step 2.1

Simplify $3 (\ln (5))$ by moving $3$ inside the logarithm.

$\frac{- \ln (1 + 9 e^{- 0.1656 t}) + \ln (5^{3}) + 3 \ln (2)}{\ln (p)}$

Step 2.2

Raise $5$ to the power of $3$ .

$\frac{- \ln (1 + 9 e^{- 0.1656 t}) + \ln (125) + 3 \ln (2)}{\ln (p)}$

Step 2.3

Simplify $3 \ln (2)$ by moving $3$ inside the logarithm.

$\frac{- \ln (1 + 9 e^{- 0.1656 t}) + \ln (125) + \ln (2^{3})}{\ln (p)}$

Step 2.4

Raise $2$ to the power of $3$ .

$\frac{- \ln (1 + 9 e^{- 0.1656 t}) + \ln (125) + \ln (8)}{\ln (p)}$

Step 3

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\frac{- \ln (1 + 9 e^{- 0.1656 t}) + \ln (125 \cdot 8)}{\ln (p)}$

Step 4

Multiply $125$ by $8$ .

$\frac{- \ln (1 + 9 e^{- 0.1656 t}) + \ln (1000)}{\ln (p)}$

Step 5

Factor $- 1$ out of $- \ln (1 + 9 e^{- 0.1656 t})$ .

$\frac{- (\ln (1 + 9 e^{- 0.1656 t})) + \ln (1000)}{\ln (p)}$

Step 6

Factor $- 1$ out of $\ln (1000)$ .

$\frac{- (\ln (1 + 9 e^{- 0.1656 t})) - 1 (- \ln (1000))}{\ln (p)}$

Step 7

Factor $- 1$ out of $- (\ln (1 + 9 e^{- 0.1656 t})) - 1 (- \ln (1000))$ .

$\frac{- (\ln (1 + 9 e^{- 0.1656 t}) - \ln (1000))}{\ln (p)}$

Step 8

Simplify the expression.

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Step 8.1

Rewrite $- (\ln (1 + 9 e^{- 0.1656 t}) - \ln (1000))$ as $- 1 (\ln (1 + 9 e^{- 0.1656 t}) - \ln (1000))$ .

$\frac{- 1 (\ln (1 + 9 e^{- 0.1656 t}) - \ln (1000))}{\ln (p)}$

Step 8.2

Move the negative in front of the fraction.

$- \frac{\ln (1 + 9 e^{- 0.1656 t}) - \ln (1000)}{\ln (p)}$