Calculus Examples

$7 \cdot 3^{t} = 5 \cdot 2^{t}$

Step 1

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

$\ln (7 \cdot 3^{t}) = \ln (5 \cdot 2^{t})$

Step 2

Expand the left side.

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Step 2.1

Rewrite $\ln (7 \cdot 3^{t})$ as $\ln (7) + \ln (3^{t})$ .

$\ln (7) + \ln (3^{t}) = \ln (5 \cdot 2^{t})$

Step 2.2

Expand $\ln (3^{t})$ by moving $t$ outside the logarithm.

$\ln (7) + t \ln (3) = \ln (5 \cdot 2^{t})$

Step 3

Expand the right side.

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Step 3.1

Rewrite $\ln (5 \cdot 2^{t})$ as $\ln (5) + \ln (2^{t})$ .

$\ln (7) + t \ln (3) = \ln (5) + \ln (2^{t})$

Step 3.2

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

$\ln (7) + t \ln (3) = \ln (5) + t \ln (2)$

Step 4

Reorder $\ln (7)$ and $t \ln (3)$ .

$t \ln (3) + \ln (7) = \ln (5) + t \ln (2)$

Step 5

Reorder $\ln (5)$ and $t \ln (2)$ .

$t \ln (3) + \ln (7) = t \ln (2) + \ln (5)$

Step 6

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

$t \ln (3) + \ln (7) - t \ln (2) - \ln (5) = 0$

Step 7

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

$t \ln (3) - t \ln (2) + \ln (\frac{7}{5}) = 0$

Step 8

Subtract $\ln (\frac{7}{5})$ from both sides of the equation.

$t \ln (3) - t \ln (2) = - \ln (\frac{7}{5})$

Step 9

Factor $t$ out of $t \ln (3) - t \ln (2)$ .

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Step 9.1

Factor $t$ out of $t \ln (3)$ .

$t (\ln (3)) - t \ln (2) = - \ln (\frac{7}{5})$

Step 9.2

Factor $t$ out of $- t \ln (2)$ .

$t (\ln (3)) + t (- 1 \ln (2)) = - \ln (\frac{7}{5})$

Step 9.3

Factor $t$ out of $t (\ln (3)) + t (- 1 \ln (2))$ .

$t (\ln (3) - 1 \ln (2)) = - \ln (\frac{7}{5})$

Step 10

Rewrite $- 1 \ln (2)$ as $- \ln (2)$ .

$t (\ln (3) - \ln (2)) = - \ln (\frac{7}{5})$

Step 11

Divide each term in $t (\ln (3) - \ln (2)) = - \ln (\frac{7}{5})$ by $\ln (3) - \ln (2)$ and simplify.

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Step 11.1

Divide each term in $t (\ln (3) - \ln (2)) = - \ln (\frac{7}{5})$ by $\ln (3) - \ln (2)$ .

$\frac{t (\ln (3) - \ln (2))}{\ln (3) - \ln (2)} = \frac{- \ln (\frac{7}{5})}{\ln (3) - \ln (2)}$

Step 11.2

Simplify the left side.

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Step 11.2.1

Cancel the common factor of $\ln (3) - \ln (2)$ .

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Step 11.2.1.1

Cancel the common factor.

$\frac{t (\ln (3) - \ln (2))}{\ln (3) - \ln (2)} = \frac{- \ln (\frac{7}{5})}{\ln (3) - \ln (2)}$

Step 11.2.1.2

Divide $t$ by $1$ .

$t = \frac{- \ln (\frac{7}{5})}{\ln (3) - \ln (2)}$

Step 11.3

Simplify the right side.

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Step 11.3.1

Move the negative in front of the fraction.

$t = - \frac{\ln (\frac{7}{5})}{\ln (3) - \ln (2)}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$t = - \frac{\ln (\frac{7}{5})}{\ln (3) - \ln (2)}$

Decimal Form:

$t = - 0.82984264 \dots$