Precalculus Examples

$- \frac{\ln (A) - \ln (1500)}{64.3775165 + 20 \ln (n) - 20 \ln (25 n + 3)}$

Step 1

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

$- \frac{\ln (\frac{A}{1500})}{64.3775165 + 20 \ln (n) - 20 \ln (25 n + 3)}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $0.0000005$ out of $64.3775165 + 20 \ln (n) - 20 \ln (25 n + 3)$ .

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Step 2.1.1

Factor $0.0000005$ out of $64.3775165$ .

$- \frac{\ln (\frac{A}{1500})}{0.0000005 (128755033) + 20 \ln (n) - 20 \ln (25 n + 3)}$

Step 2.1.2

Factor $0.0000005$ out of $20 \ln (n)$ .

$- \frac{\ln (\frac{A}{1500})}{0.0000005 (128755033) + 0.0000005 (40000000 \ln (n)) - 20 \ln (25 n + 3)}$

Step 2.1.3

Factor $0.0000005$ out of $- 20 \ln (25 n + 3)$ .

$- \frac{\ln (\frac{A}{1500})}{0.0000005 (128755033) + 0.0000005 (40000000 \ln (n)) + 0.0000005 (- 40000000 \ln (25 n + 3))}$

Step 2.1.4

Factor $0.0000005$ out of $0.0000005 (128755033) + 0.0000005 (40000000 \ln (n))$ .

$- \frac{\ln (\frac{A}{1500})}{0.0000005 (128755033 + 40000000 \ln (n)) + 0.0000005 (- 40000000 \ln (25 n + 3))}$

Step 2.1.5

Factor $0.0000005$ out of $0.0000005 (128755033 + 40000000 \ln (n)) + 0.0000005 (- 40000000 \ln (25 n + 3))$ .

$- \frac{\ln (\frac{A}{1500})}{0.0000005 (128755033 + 40000000 \ln (n) - 40000000 \ln (25 n + 3))}$

Step 2.2

Simplify $40000000 \ln (n)$ by moving $40000000$ inside the logarithm.

$- \frac{\ln (\frac{A}{1500})}{0.0000005 (128755033 + \ln (n^{40000000}) - 40000000 \ln (25 n + 3))}$

Step 2.3

Simplify $- 40000000 \ln (25 n + 3)$ by moving $40000000$ inside the logarithm.

$- \frac{\ln (\frac{A}{1500})}{0.0000005 (128755033 + \ln (n^{40000000}) - \ln ({(25 n + 3)}^{40000000}))}$

Step 2.4

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

$- \frac{\ln (\frac{A}{1500})}{0.0000005 (128755033 + \ln (\frac{n^{40000000}}{{(25 n + 3)}^{40000000}}))}$

Step 3

Multiply by $1$ .

$- \frac{1 \ln (\frac{A}{1500})}{0.0000005 (128755033 + \ln (\frac{n^{40000000}}{{(25 n + 3)}^{40000000}}))}$

Step 4

Separate fractions.

$- (\frac{1}{0.0000005} \cdot \frac{\ln (\frac{A}{1500})}{128755033 + \ln (\frac{n^{40000000}}{{(25 n + 3)}^{40000000}})})$

Step 5

Divide $1$ by $0.0000005$ .

$- (2000000 \frac{\ln (\frac{A}{1500})}{128755033 + \ln (\frac{n^{40000000}}{{(25 n + 3)}^{40000000}})})$

Step 6

Multiply $2000000 \frac{\ln (\frac{A}{1500})}{128755033 + \ln (\frac{n^{40000000}}{{(25 n + 3)}^{40000000}})}$ .

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Step 6.1

Combine $2000000$ and $\frac{\ln (\frac{A}{1500})}{128755033 + \ln (\frac{n^{40000000}}{{(25 n + 3)}^{40000000}})}$ .

$- \frac{2000000 \ln (\frac{A}{1500})}{128755033 + \ln (\frac{n^{40000000}}{{(25 n + 3)}^{40000000}})}$

Step 6.2

Simplify $2000000 \ln (\frac{A}{1500})$ by moving $2000000$ inside the logarithm.

$- \frac{\ln ({(\frac{A}{1500})}^{2000000})}{128755033 + \ln (\frac{n^{40000000}}{{(25 n + 3)}^{40000000}})}$

Step 7

Apply the product rule to $\frac{A}{1500}$ .

$- \frac{\ln (\frac{A^{2000000}}{1500^{2000000}})}{128755033 + \ln (\frac{n^{40000000}}{{(25 n + 3)}^{40000000}})}$