Precalculus Examples

$3 (2^{2 t - 5}) - 4 = 10$

Step 1

Move all terms not containing $t$ to the right side of the equation.

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Step 1.1

Add $4$ to both sides of the equation.

$3 \cdot 2^{2 t - 5} = 10 + 4$

Step 1.2

Add $10$ and $4$ .

$3 \cdot 2^{2 t - 5} = 14$

Step 2

Divide each term in $3 \cdot 2^{2 t - 5} = 14$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 \cdot 2^{2 t - 5} = 14$ by $3$ .

$\frac{3 \cdot 2^{2 t - 5}}{3} = \frac{14}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 \cdot 2^{2 t - 5}}{3} = \frac{14}{3}$

Step 2.2.1.2

Divide $2^{2 t - 5}$ by $1$ .

$2^{2 t - 5} = \frac{14}{3}$

Step 3

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

$\ln (2^{2 t - 5}) = \ln (\frac{14}{3})$

Step 4

Expand $\ln (2^{2 t - 5})$ by moving $2 t - 5$ outside the logarithm.

$(2 t - 5) \ln (2) = \ln (\frac{14}{3})$

Step 5

Simplify the left side.

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Step 5.1

Apply the distributive property.

$2 t \ln (2) - 5 \ln (2) = \ln (\frac{14}{3})$

Step 6

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

$2 t \ln (2) - 5 \ln (2) - \ln (\frac{14}{3}) = 0$

Step 7

Move all terms not containing $t$ to the right side of the equation.

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Step 7.1

Add $5 \ln (2)$ to both sides of the equation.

$2 t \ln (2) - \ln (\frac{14}{3}) = 5 \ln (2)$

Step 7.2

Add $\ln (\frac{14}{3})$ to both sides of the equation.

$2 t \ln (2) = 5 \ln (2) + \ln (\frac{14}{3})$

Step 8

Divide each term in $2 t \ln (2) = 5 \ln (2) + \ln (\frac{14}{3})$ by $2 \ln (2)$ and simplify.

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Step 8.1

Divide each term in $2 t \ln (2) = 5 \ln (2) + \ln (\frac{14}{3})$ by $2 \ln (2)$ .

$\frac{2 t \ln (2)}{2 \ln (2)} = \frac{5 \ln (2)}{2 \ln (2)} + \frac{\ln (\frac{14}{3})}{2 \ln (2)}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $2$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{2 t \ln (2)}{2 \ln (2)} = \frac{5 \ln (2)}{2 \ln (2)} + \frac{\ln (\frac{14}{3})}{2 \ln (2)}$

Step 8.2.1.2

Rewrite the expression.

$\frac{t \ln (2)}{\ln (2)} = \frac{5 \ln (2)}{2 \ln (2)} + \frac{\ln (\frac{14}{3})}{2 \ln (2)}$

Step 8.2.2

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

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Step 8.2.2.1

Cancel the common factor.

$\frac{t \ln (2)}{\ln (2)} = \frac{5 \ln (2)}{2 \ln (2)} + \frac{\ln (\frac{14}{3})}{2 \ln (2)}$

Step 8.2.2.2

Divide $t$ by $1$ .

$t = \frac{5 \ln (2)}{2 \ln (2)} + \frac{\ln (\frac{14}{3})}{2 \ln (2)}$

Step 8.3

Simplify the right side.

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Step 8.3.1

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

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Step 8.3.1.1

Cancel the common factor.

$t = \frac{5 \ln (2)}{2 \ln (2)} + \frac{\ln (\frac{14}{3})}{2 \ln (2)}$

Step 8.3.1.2

Rewrite the expression.

$t = \frac{5}{2} + \frac{\ln (\frac{14}{3})}{2 \ln (2)}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$t = \frac{5}{2} + \frac{\ln (\frac{14}{3})}{2 \ln (2)}$

Decimal Form:

$t = 3.61119621 \dots$