Finite Math Examples

$3 {(9)}^{x - 3} = 25$

Step 1

Subtract $25$ from both sides of the equation.

$3 {(9)}^{x - 3} - 25 = 0$

Step 2

Multiply $3 \cdot 9^{x - 3}$ .

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Step 2.1

Rewrite $9$ as $3^{2}$ .

$3 \cdot {(3^{2})}^{x - 3} - 25 = 0$

Step 2.2

Multiply the exponents in ${(3^{2})}^{x - 3}$ .

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Step 2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3 \cdot 3^{2 (x - 3)} - 25 = 0$

Step 2.2.2

Apply the distributive property.

$3 \cdot 3^{2 x + 2 \cdot - 3} - 25 = 0$

Step 2.2.3

Multiply $2$ by $- 3$ .

$3 \cdot 3^{2 x - 6} - 25 = 0$

Step 2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3^{1 + 2 x - 6} - 25 = 0$

Step 2.4

Subtract $6$ from $1$ .

$3^{2 x - 5} - 25 = 0$

Step 3

Add $25$ to both sides of the equation.

$3^{2 x - 5} = 25$

Step 4

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

$\ln (3^{2 x - 5}) = \ln (25)$

Step 5

Expand $\ln (3^{2 x - 5})$ by moving $2 x - 5$ outside the logarithm.

$(2 x - 5) \ln (3) = \ln (25)$

Step 6

Simplify the left side.

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Step 6.1

Apply the distributive property.

$2 x \ln (3) - 5 \ln (3) = \ln (25)$

Step 7

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

$2 x \ln (3) - 5 \ln (3) - \ln (25) = 0$

Step 8

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

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Step 8.1

Add $5 \ln (3)$ to both sides of the equation.

$2 x \ln (3) - \ln (25) = 5 \ln (3)$

Step 8.2

Add $\ln (25)$ to both sides of the equation.

$2 x \ln (3) = 5 \ln (3) + \ln (25)$

Step 9

Divide each term in $2 x \ln (3) = 5 \ln (3) + \ln (25)$ by $2 \ln (3)$ and simplify.

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Step 9.1

Divide each term in $2 x \ln (3) = 5 \ln (3) + \ln (25)$ by $2 \ln (3)$ .

$\frac{2 x \ln (3)}{2 \ln (3)} = \frac{5 \ln (3)}{2 \ln (3)} + \frac{\ln (25)}{2 \ln (3)}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $2$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{2 x \ln (3)}{2 \ln (3)} = \frac{5 \ln (3)}{2 \ln (3)} + \frac{\ln (25)}{2 \ln (3)}$

Step 9.2.1.2

Rewrite the expression.

$\frac{x \ln (3)}{\ln (3)} = \frac{5 \ln (3)}{2 \ln (3)} + \frac{\ln (25)}{2 \ln (3)}$

Step 9.2.2

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

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Step 9.2.2.1

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{5 \ln (3)}{2 \ln (3)} + \frac{\ln (25)}{2 \ln (3)}$

Step 9.2.2.2

Divide $x$ by $1$ .

$x = \frac{5 \ln (3)}{2 \ln (3)} + \frac{\ln (25)}{2 \ln (3)}$

Step 9.3

Simplify the right side.

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Step 9.3.1

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

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Step 9.3.1.1

Cancel the common factor.

$x = \frac{5 \ln (3)}{2 \ln (3)} + \frac{\ln (25)}{2 \ln (3)}$

Step 9.3.1.2

Rewrite the expression.

$x = \frac{5}{2} + \frac{\ln (25)}{2 \ln (3)}$

Step 10

Simplify each term.

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Step 10.1

Rewrite $\ln (25)$ as $\ln (5^{2})$ .

$x = \frac{5}{2} + \frac{\ln (5^{2})}{2 \ln (3)}$

Step 10.2

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

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

Step 10.3

Cancel the common factor of $2$ .

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Step 10.3.1

Cancel the common factor.

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

Step 10.3.2

Rewrite the expression.

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