Calculus Examples

${(\frac{1}{25})}^{1 - 3 x^{2}} = 3^{4 x}$

Step 1

Take the log of both sides of the equation.

$\ln ({(\frac{1}{25})}^{1 - 3 x^{2}}) = \ln (3^{4 x})$

Step 2

Expand $\ln ({(\frac{1}{25})}^{1 - 3 x^{2}})$ by moving $1 - 3 x^{2}$ outside the logarithm.

$(1 - 3 x^{2}) \ln (\frac{1}{25}) = \ln (3^{4 x})$

Step 3

Rewrite $\ln (\frac{1}{25})$ as $\ln (1) - \ln (25)$ .

$(1 - 3 x^{2}) (\ln (1) - \ln (25)) = \ln (3^{4 x})$

Step 4

The natural logarithm of $1$ is $0$ .

$(1 - 3 x^{2}) (0 - \ln (25)) = \ln (3^{4 x})$

Step 5

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

$(1 - 3 x^{2}) (0 - \ln (5^{2})) = \ln (3^{4 x})$

Step 6

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

$(1 - 3 x^{2}) (0 - (2 \ln (5))) = \ln (3^{4 x})$

Step 7

Multiply $2$ by $- 1$ .

$(1 - 3 x^{2}) (0 - 2 \ln (5)) = \ln (3^{4 x})$

Step 8

Subtract $2 \ln (5)$ from $0$ .

$(1 - 3 x^{2}) (- 2 \ln (5)) = \ln (3^{4 x})$

Step 9

Expand $\ln (3^{4 x})$ by moving $4 x$ outside the logarithm.

$(1 - 3 x^{2}) (- 2 \ln (5)) = 4 x \ln (3)$

Step 10

Solve the equation for $x$ .

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Step 10.1

Simplify $(1 - 3 x^{2}) (- 2 \ln (5))$ .

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Step 10.1.1

Rewrite.

$0 + 0 + (1 - 3 x^{2}) (- 2 \ln (5)) = 4 x \ln (3)$

Step 10.1.2

Simplify by adding zeros.

$(1 - 3 x^{2}) (- 2 \ln (5)) = 4 x \ln (3)$

Step 10.1.3

Apply the distributive property.

$1 (- 2 \ln (5)) - 3 x^{2} (- 2 \ln (5)) = 4 x \ln (3)$

Step 10.1.4

Simplify the expression.

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Step 10.1.4.1

Multiply $- 2 \ln (5)$ by $1$ .

$- 2 \ln (5) - 3 x^{2} (- 2 \ln (5)) = 4 x \ln (3)$

Step 10.1.4.2

Rewrite using the commutative property of multiplication.

$- 2 \ln (5) - 3 \cdot - 2 x^{2} \ln (5) = 4 x \ln (3)$

Step 10.1.4.3

Multiply $- 3$ by $- 2$ .

$- 2 \ln (5) + 6 x^{2} \ln (5) = 4 x \ln (3)$

Step 10.2

Subtract $4 x \ln (3)$ from both sides of the equation.

$- 2 \ln (5) + 6 x^{2} \ln (5) - 4 x \ln (3) = 0$

Step 10.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 10.4

Substitute the values $a = 6 \ln (5)$ , $b = - 4 \ln (3)$ , and $c = - 2 \ln (5)$ into the quadratic formula and solve for $x$ .

$\frac{4 \ln (3) \pm \sqrt{{(- 4 \ln (3))}^{2} - 4 \cdot (6 \ln (5) \cdot (- 2 \ln (5)))}}{2 (6 \ln (5))}$

Step 10.5

Simplify.

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Step 10.5.1

Simplify the numerator.

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Step 10.5.1.1

Add parentheses.

$x = \frac{4 \ln (3) \pm \sqrt{{(- 4 \ln (3))}^{2} - 4 \cdot (6 (\ln (5) \cdot (- 2 \ln (5))))}}{2 \cdot (6 \ln (5))}$

Step 10.5.1.2

Let $u = 6 (\ln (5) \cdot (- 2 \ln (5)))$ . Substitute $u$ for all occurrences of $6 (\ln (5) \cdot (- 2 \ln (5)))$ .

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Step 10.5.1.2.1

Apply the product rule to $- 4 \ln (3)$ .

$x = \frac{4 \ln (3) \pm \sqrt{{(- 4)}^{2} \ln^{2} (3) - 4 \cdot u}}{2 \cdot (6 \ln (5))}$

Step 10.5.1.2.2

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \ln (3) \pm \sqrt{16 \ln^{2} (3) - 4 u}}{2 \cdot (6 \ln (5))}$

Step 10.5.1.3

Factor $4$ out of $16 \ln^{2} (3) - 4 u$ .

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Step 10.5.1.3.1

Factor $4$ out of $16 \ln^{2} (3)$ .

$x = \frac{4 \ln (3) \pm \sqrt{4 (4 \ln^{2} (3)) - 4 u}}{2 \cdot (6 \ln (5))}$

Step 10.5.1.3.2

Factor $4$ out of $- 4 u$ .

$x = \frac{4 \ln (3) \pm \sqrt{4 (4 \ln^{2} (3)) + 4 (- u)}}{2 \cdot (6 \ln (5))}$

Step 10.5.1.3.3

Factor $4$ out of $4 (4 \ln^{2} (3)) + 4 (- u)$ .

$x = \frac{4 \ln (3) \pm \sqrt{4 (4 \ln^{2} (3) - u)}}{2 \cdot (6 \ln (5))}$

Step 10.5.1.4

Replace all occurrences of $u$ with $6 (\ln (5) \cdot (- 2 \ln (5)))$ .

$x = \frac{4 \ln (3) \pm \sqrt{4 (4 \ln^{2} (3) - (6 (\ln (5) \cdot (- 2 \ln (5)))))}}{2 \cdot (6 \ln (5))}$

Step 10.5.1.5

Simplify each term.

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Step 10.5.1.5.1

Multiply $\ln (5)$ by $\ln (5)$ by adding the exponents.

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Step 10.5.1.5.1.1

Move $\ln (5)$ .

$x = \frac{4 \ln (3) \pm \sqrt{4 (4 \ln^{2} (3) - (6 (\ln (5) \ln (5) \cdot - 2)))}}{2 \cdot (6 \ln (5))}$

Step 10.5.1.5.1.2

Multiply $\ln (5)$ by $\ln (5)$ .

$x = \frac{4 \ln (3) \pm \sqrt{4 (4 \ln^{2} (3) - (6 (\ln^{2} (5) \cdot - 2)))}}{2 \cdot (6 \ln (5))}$

Step 10.5.1.5.2

Move $- 2$ to the left of $\ln^{2} (5)$ .

$x = \frac{4 \ln (3) \pm \sqrt{4 (4 \ln^{2} (3) - (6 (- 2 \cdot \ln^{2} (5))))}}{2 \cdot (6 \ln (5))}$

Step 10.5.1.5.3

Multiply $- 2$ by $6$ .

$x = \frac{4 \ln (3) \pm \sqrt{4 (4 \ln^{2} (3) - (- 12 \ln^{2} (5)))}}{2 \cdot (6 \ln (5))}$

Step 10.5.1.5.4

Multiply $- 12$ by $- 1$ .

$x = \frac{4 \ln (3) \pm \sqrt{4 (4 \ln^{2} (3) + 12 \ln^{2} (5))}}{2 \cdot (6 \ln (5))}$

Step 10.5.1.6

Factor $4$ out of $4 \ln^{2} (3) + 12 \ln^{2} (5)$ .

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Step 10.5.1.6.1

Factor $4$ out of $4 \ln^{2} (3)$ .

$x = \frac{4 \ln (3) \pm \sqrt{4 (4 \ln^{2} (3) + 12 \ln^{2} (5))}}{2 \cdot (6 \ln (5))}$

Step 10.5.1.6.2

Factor $4$ out of $12 \ln^{2} (5)$ .

$x = \frac{4 \ln (3) \pm \sqrt{4 (4 \ln^{2} (3) + 4 (3 \ln^{2} (5)))}}{2 \cdot (6 \ln (5))}$

Step 10.5.1.6.3

Factor $4$ out of $4 \ln^{2} (3) + 4 (3 \ln^{2} (5))$ .

$x = \frac{4 \ln (3) \pm \sqrt{4 (4 (\ln^{2} (3) + 3 \ln^{2} (5)))}}{2 \cdot (6 \ln (5))}$

$x = \frac{4 \ln (3) \pm \sqrt{4 \cdot (4 (\ln^{2} (3) + 3 \ln^{2} (5)))}}{2 \cdot (6 \ln (5))}$

Step 10.5.1.7

Multiply $4$ by $4$ .

$x = \frac{4 \ln (3) \pm \sqrt{16 (\ln^{2} (3) + 3 \ln^{2} (5))}}{2 \cdot (6 \ln (5))}$

Step 10.5.1.8

Rewrite $16 (\ln^{2} (3) + 3 \ln^{2} (5))$ as ${(2^{2})}^{2} (\ln^{2} (3) + 3 \ln^{2} (5))$ .

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Step 10.5.1.8.1

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

$x = \frac{4 \ln (3) \pm \sqrt{4^{2} (\ln^{2} (3) + 3 \ln^{2} (5))}}{2 \cdot (6 \ln (5))}$

Step 10.5.1.8.2

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

$x = \frac{4 \ln (3) \pm \sqrt{{(2^{2})}^{2} (\ln^{2} (3) + 3 \ln^{2} (5))}}{2 \cdot (6 \ln (5))}$

Step 10.5.1.9

Pull terms out from under the radical.

$x = \frac{4 \ln (3) \pm 2^{2} \sqrt{\ln^{2} (3) + 3 \ln^{2} (5)}}{2 \cdot (6 \ln (5))}$

Step 10.5.1.10

Raise $2$ to the power of $2$ .

$x = \frac{4 \ln (3) \pm 4 \sqrt{\ln^{2} (3) + 3 \ln^{2} (5)}}{2 \cdot (6 \ln (5))}$

Step 10.5.2

Multiply $2$ by $6$ .

$x = \frac{4 \ln (3) \pm 4 \sqrt{\ln^{2} (3) + 3 \ln^{2} (5)}}{12 \ln (5)}$

Step 10.5.3

Simplify $\frac{4 \ln (3) \pm 4 \sqrt{\ln^{2} (3) + 3 \ln^{2} (5)}}{12 \ln (5)}$ .

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

Step 10.6

The final answer is the combination of both solutions.

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

Step 11

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.84810424 \dots, - 0.39303344 \dots$