Finite Math Examples

$3^{t - 1} = 7^{t - 1}$

Step 1

Take the log of both sides of the equation.

$\ln (3^{t - 1}) = \ln (7^{t - 1})$

Step 2

Expand $\ln (3^{t - 1})$ by moving $t - 1$ outside the logarithm.

$(t - 1) \ln (3) = \ln (7^{t - 1})$

Step 3

Expand $\ln (7^{t - 1})$ by moving $t - 1$ outside the logarithm.

$(t - 1) \ln (3) = (t - 1) \ln (7)$

Step 4

Solve the equation for $t$ .

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Step 4.1

Simplify $(t - 1) \ln (3)$ .

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Step 4.1.1

Rewrite.

$0 + 0 + (t - 1) \ln (3) = (t - 1) \ln (7)$

Step 4.1.2

Simplify by adding zeros.

$(t - 1) \ln (3) = (t - 1) \ln (7)$

Step 4.1.3

Apply the distributive property.

$t \ln (3) - 1 \ln (3) = (t - 1) \ln (7)$

Step 4.1.4

Rewrite $- 1 \ln (3)$ as $- \ln (3)$ .

$t \ln (3) - \ln (3) = (t - 1) \ln (7)$

Step 4.2

Simplify $(t - 1) \ln (7)$ .

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Step 4.2.1

Apply the distributive property.

$t \ln (3) - \ln (3) = t \ln (7) - 1 \ln (7)$

Step 4.2.2

Rewrite $- 1 \ln (7)$ as $- \ln (7)$ .

$t \ln (3) - \ln (3) = t \ln (7) - \ln (7)$

Step 4.3

Subtract $t \ln (7)$ from both sides of the equation.

$t \ln (3) - \ln (3) - t \ln (7) = - \ln (7)$

Step 4.4

Add $\ln (3)$ to both sides of the equation.

$t \ln (3) - t \ln (7) = - \ln (7) + \ln (3)$

Step 4.5

Factor $t$ out of $t \ln (3) - t \ln (7)$ .

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Step 4.5.1

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

$t (\ln (3)) - t \ln (7) = - \ln (7) + \ln (3)$

Step 4.5.2

Factor $t$ out of $- t \ln (7)$ .

$t (\ln (3)) + t (- 1 \ln (7)) = - \ln (7) + \ln (3)$

Step 4.5.3

Factor $t$ out of $t (\ln (3)) + t (- 1 \ln (7))$ .

$t (\ln (3) - 1 \ln (7)) = - \ln (7) + \ln (3)$

Step 4.6

Rewrite $- 1 \ln (7)$ as $- \ln (7)$ .

$t (\ln (3) - \ln (7)) = - \ln (7) + \ln (3)$

Step 4.7

Divide each term in $t (\ln (3) - \ln (7)) = - \ln (7) + \ln (3)$ by $\ln (3) - \ln (7)$ and simplify.

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Step 4.7.1

Divide each term in $t (\ln (3) - \ln (7)) = - \ln (7) + \ln (3)$ by $\ln (3) - \ln (7)$ .

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

Step 4.7.2

Simplify the left side.

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Step 4.7.2.1

Cancel the common factor of $\ln (3) - \ln (7)$ .

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Step 4.7.2.1.1

Cancel the common factor.

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

Step 4.7.2.1.2

Divide $t$ by $1$ .

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

Step 4.7.3

Simplify the right side.

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Step 4.7.3.1

Combine the numerators over the common denominator.

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

Step 4.7.3.2

Cancel the common factor of $- \ln (7) + \ln (3)$ and $\ln (3) - \ln (7)$ .

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Step 4.7.3.2.1

Reorder terms.

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

Step 4.7.3.2.2

Cancel the common factor.

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

Step 4.7.3.2.3

Rewrite the expression.

$t = 1$