Precalculus Examples

$25^{x} - 8 \cdot 5^{x} = - 16$

Step 1

Add $16$ to both sides of the equation.

$25^{x} - 8 \cdot 5^{x} + 16 = 0$

Step 2

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

${(5^{2})}^{x} - 8 \cdot 5^{x} + 16 = 0$

Step 3

Rewrite ${(5^{2})}^{x}$ as ${(5^{x})}^{2}$ .

${(5^{x})}^{2} - 8 \cdot 5^{x} + 16 = 0$

Step 4

Let $u = 5^{x}$ . Substitute $u$ for all occurrences of $5^{x}$ .

$u^{2} - 8 u + 16 = 0$

Step 5

Factor using the perfect square rule.

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Step 5.1

Rewrite $16$ as $4^{2}$ .

$u^{2} - 8 u + 4^{2} = 0$

Step 5.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 u = 2 \cdot u \cdot 4$

Step 5.3

Rewrite the polynomial.

$u^{2} - 2 \cdot u \cdot 4 + 4^{2} = 0$

Step 5.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 4$ .

${(u - 4)}^{2} = 0$

Step 6

Replace all occurrences of $u$ with $5^{x}$ .

${(5^{x} - 4)}^{2} = 0$

Step 7

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$5^{x} - 4 = \pm \sqrt{0}$

Step 8

Simplify $\pm \sqrt{0}$ .

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Step 8.1

Rewrite $0$ as $0^{2}$ .

$5^{x} - 4 = \pm \sqrt{0^{2}}$

Step 8.2

Pull terms out from under the radical, assuming positive real numbers.

$5^{x} - 4 = \pm 0$

Step 8.3

Plus or minus $0$ is $0$ .

$5^{x} - 4 = 0$

Step 9

Add $4$ to both sides of the equation.

$5^{x} = 4$

Step 10

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

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

Step 11

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

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

Step 12

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

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Step 12.1

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

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

Step 12.2

Simplify the left side.

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Step 12.2.1

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

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Step 12.2.1.1

Cancel the common factor.

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

Step 12.2.1.2

Divide $x$ by $1$ .

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