Finite Math Examples

$5^{2 x} + 3 (5^{x}) = 28$

Step 1

Rewrite $5^{2 x}$ as exponentiation.

${(5^{x})}^{2} + 3 \cdot 5^{x} = 28$

Step 2

Substitute $u$ for $5^{x}$ .

$u^{2} + 3 u = 28$

Step 3

Solve for $u$ .

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Step 3.1

Subtract $28$ from both sides of the equation.

$u^{2} + 3 u - 28 = 0$

Step 3.2

Factor $u^{2} + 3 u - 28$ using the AC method.

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Step 3.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 28$ and whose sum is $3$ .

$- 4, 7$

Step 3.2.2

Write the factored form using these integers.

$(u - 4) (u + 7) = 0$

Step 3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$u - 4 = 0$

$u + 7 = 0$

Step 3.4

Set $u - 4$ equal to $0$ and solve for $u$ .

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Step 3.4.1

Set $u - 4$ equal to $0$ .

$u - 4 = 0$

Step 3.4.2

Add $4$ to both sides of the equation.

$u = 4$

Step 3.5

Set $u + 7$ equal to $0$ and solve for $u$ .

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Step 3.5.1

Set $u + 7$ equal to $0$ .

$u + 7 = 0$

Step 3.5.2

Subtract $7$ from both sides of the equation.

$u = - 7$

Step 3.6

The final solution is all the values that make $(u - 4) (u + 7) = 0$ true.

$u = 4, - 7$

Step 4

Substitute $4$ for $u$ in $u = 5^{x}$ .

$4 = 5^{x}$

Step 5

Solve $4 = 5^{x}$ .

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Step 5.1

Rewrite the equation as $5^{x} = 4$ .

$5^{x} = 4$

Step 5.2

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

$\ln (5^{x}) = \ln (4)$

Step 5.3

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

$x \ln (5) = \ln (4)$

Step 5.4

Divide each term in $x \ln (5) = \ln (4)$ by $\ln (5)$ and simplify.

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Step 5.4.1

Divide each term in $x \ln (5) = \ln (4)$ by $\ln (5)$ .

$\frac{x \ln (5)}{\ln (5)} = \frac{\ln (4)}{\ln (5)}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $\ln (5)$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{x \ln (5)}{\ln (5)} = \frac{\ln (4)}{\ln (5)}$

Step 5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (4)}{\ln (5)}$

Step 6

Substitute $- 7$ for $u$ in $u = 5^{x}$ .

$- 7 = 5^{x}$

Step 7

Solve $- 7 = 5^{x}$ .

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Step 7.1

Rewrite the equation as $5^{x} = - 7$ .

$5^{x} = - 7$

Step 7.2

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

$\ln (5^{x}) = \ln (- 7)$

Step 7.3

The equation cannot be solved because $\ln (- 7)$ is undefined.

Undefined

Step 7.4

There is no solution for $5^{x} = - 7$

No solution

Step 8

List the solutions that makes the equation true.

$x = \frac{\ln (4)}{\ln (5)}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (4)}{\ln (5)}$

Decimal Form:

$x = 0.86135311 \dots$