Precalculus Examples

${(0.25)}^{5} = 4^{\frac{5 x - 3}{3}} {(0.125)}^{6 x}$

Step 1

Rewrite the equation as $4^{\frac{5 x - 3}{3}} \cdot {0.125}^{6 x} = {0.25}^{5}$ .

$4^{\frac{5 x - 3}{3}} \cdot {0.125}^{6 x} = {0.25}^{5}$

Step 2

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

$\ln (4^{\frac{5 x - 3}{3}} \cdot {0.125}^{6 x}) = \ln ({0.25}^{5})$

Step 3

Expand the left side.

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Step 3.1

Rewrite $\ln (4^{\frac{5 x - 3}{3}} \cdot {0.125}^{6 x})$ as $\ln (4^{\frac{5 x - 3}{3}}) + \ln ({0.125}^{6 x})$ .

$\ln (4^{\frac{5 x - 3}{3}}) + \ln ({0.125}^{6 x}) = \ln ({0.25}^{5})$

Step 3.2

Expand $\ln (4^{\frac{5 x - 3}{3}})$ by moving $\frac{5 x - 3}{3}$ outside the logarithm.

$\frac{5 x - 3}{3} \ln (4) + \ln ({0.125}^{6 x}) = \ln ({0.25}^{5})$

Step 3.3

Expand $\ln ({0.125}^{6 x})$ by moving $6 x$ outside the logarithm.

$\frac{5 x - 3}{3} \ln (4) + 6 x \ln (0.125) = \ln ({0.25}^{5})$

Step 3.4

Combine $\frac{5 x - 3}{3}$ and $\ln (4)$ .

$\frac{(5 x - 3) \ln (4)}{3} + 6 x \ln (0.125) = \ln ({0.25}^{5})$

Step 4

Simplify the left side.

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Step 4.1

Reorder factors in $\frac{(5 x - 3) \ln (4)}{3} + 6 x \ln (0.125)$ .

$\frac{\ln (4) (5 x - 3)}{3} + 6 x \ln (0.125) = \ln ({0.25}^{5})$

Step 5

Simplify the right side.

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Step 5.1

Raise $0.25$ to the power of $5$ .

$\frac{\ln (4) (5 x - 3)}{3} + 6 x \ln (0.125) = \ln (0.00097656)$

Step 6

Simplify the left side.

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Step 6.1

To write $6 x \ln (0.125)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\ln (4) (5 x - 3)}{3} + 6 x \ln (0.125) \cdot \frac{3}{3} = \ln (0.00097656)$

Step 6.2

Simplify terms.

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Step 6.2.1

Combine $6 x \ln (0.125)$ and $\frac{3}{3}$ .

$\frac{\ln (4) (5 x - 3)}{3} + \frac{6 x \ln (0.125) \cdot 3}{3} = \ln (0.00097656)$

Step 6.2.2

Combine the numerators over the common denominator.

$\frac{\ln (4) (5 x - 3) + 6 x \ln (0.125) \cdot 3}{3} = \ln (0.00097656)$

Step 6.3

Simplify the numerator.

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Step 6.3.1

Apply the distributive property.

$\frac{\ln (4) (5 x) + \ln (4) \cdot - 3 + 6 x \ln (0.125) \cdot 3}{3} = \ln (0.00097656)$

Step 6.3.2

Move $- 3$ to the left of $\ln (4)$ .

$\frac{\ln (4) \cdot (5 x) - 3 \cdot \ln (4) + 6 x \ln (0.125) \cdot 3}{3} = \ln (0.00097656)$

Step 6.3.3

Move $5$ to the left of $\ln (4)$ .

$\frac{5 \ln (4) x - 3 \ln (4) + 6 x \ln (0.125) \cdot 3}{3} = \ln (0.00097656)$

Step 6.3.4

Multiply $3$ by $6$ .

$\frac{5 \ln (4) x - 3 \ln (4) + 18 x \ln (0.125)}{3} = \ln (0.00097656)$

Step 6.3.5

Reorder terms.

$\frac{5 x \ln (4) + 18 x \ln (0.125) - 3 \ln (4)}{3} = \ln (0.00097656)$

Step 7

Multiply both sides by $3$ .

$\frac{5 x \ln (4) + 18 x \ln (0.125) - 3 \ln (4)}{3} \cdot 3 = \ln (0.00097656) \cdot 3$

Step 8

Simplify.

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Step 8.1

Simplify the left side.

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Step 8.1.1

Simplify $\frac{5 x \ln (4) + 18 x \ln (0.125) - 3 \ln (4)}{3} \cdot 3$ .

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Step 8.1.1.1

Cancel the common factor of $3$ .

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Step 8.1.1.1.1

Cancel the common factor.

$\frac{5 x \ln (4) + 18 x \ln (0.125) - 3 \ln (4)}{3} \cdot 3 = \ln (0.00097656) \cdot 3$

Step 8.1.1.1.2

Rewrite the expression.

$5 x \ln (4) + 18 x \ln (0.125) - 3 \ln (4) = \ln (0.00097656) \cdot 3$

Step 8.1.1.2

Reorder $5 x \ln (4)$ and $18 x \ln (0.125)$ .

$18 x \ln (0.125) + 5 x \ln (4) - 3 \ln (4) = \ln (0.00097656) \cdot 3$

Step 8.2

Simplify the right side.

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Step 8.2.1

Move $3$ to the left of $\ln (0.00097656)$ .

$18 x \ln (0.125) + 5 x \ln (4) - 3 \ln (4) = 3 \ln (0.00097656)$

Step 9

Solve for $x$ .

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Step 9.1

Add $3 \ln (4)$ to both sides of the equation.

$18 x \ln (0.125) + 5 x \ln (4) = 3 \ln (0.00097656) + 3 \ln (4)$

Step 9.2

Factor $x$ out of $18 x \ln (0.125) + 5 x \ln (4)$ .

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Step 9.2.1

Factor $x$ out of $18 x \ln (0.125)$ .

$x (18 \ln (0.125)) + 5 x \ln (4) = 3 \ln (0.00097656) + 3 \ln (4)$

Step 9.2.2

Factor $x$ out of $5 x \ln (4)$ .

$x (18 \ln (0.125)) + x (5 \ln (4)) = 3 \ln (0.00097656) + 3 \ln (4)$

Step 9.2.3

Factor $x$ out of $x (18 \ln (0.125)) + x (5 \ln (4))$ .

$x (18 \ln (0.125) + 5 \ln (4)) = 3 \ln (0.00097656) + 3 \ln (4)$

Step 9.3

Divide each term in $x (18 \ln (0.125) + 5 \ln (4)) = 3 \ln (0.00097656) + 3 \ln (4)$ by $18 \ln (0.125) + 5 \ln (4)$ and simplify.

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Step 9.3.1

Divide each term in $x (18 \ln (0.125) + 5 \ln (4)) = 3 \ln (0.00097656) + 3 \ln (4)$ by $18 \ln (0.125) + 5 \ln (4)$ .

$\frac{x (18 \ln (0.125) + 5 \ln (4))}{18 \ln (0.125) + 5 \ln (4)} = \frac{3 \ln (0.00097656)}{18 \ln (0.125) + 5 \ln (4)} + \frac{3 \ln (4)}{18 \ln (0.125) + 5 \ln (4)}$

Step 9.3.2

Simplify the left side.

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Step 9.3.2.1

Cancel the common factor of $18 \ln (0.125) + 5 \ln (4)$ .

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Step 9.3.2.1.1

Cancel the common factor.

$\frac{x (18 \ln (0.125) + 5 \ln (4))}{18 \ln (0.125) + 5 \ln (4)} = \frac{3 \ln (0.00097656)}{18 \ln (0.125) + 5 \ln (4)} + \frac{3 \ln (4)}{18 \ln (0.125) + 5 \ln (4)}$

Step 9.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{3 \ln (0.00097656)}{18 \ln (0.125) + 5 \ln (4)} + \frac{3 \ln (4)}{18 \ln (0.125) + 5 \ln (4)}$

Step 9.3.3

Simplify the right side.

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Step 9.3.3.1

Combine the numerators over the common denominator.

$x = \frac{3 \ln (0.00097656) + 3 \ln (4)}{18 \ln (0.125) + 5 \ln (4)}$

Step 9.3.3.2

Factor $3$ out of $3 \ln (0.00097656) + 3 \ln (4)$ .

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Step 9.3.3.2.1

Factor $3$ out of $3 \ln (0.00097656)$ .

$x = \frac{3 \ln (0.00097656) + 3 \ln (4)}{18 \ln (0.125) + 5 \ln (4)}$

Step 9.3.3.2.2

Factor $3$ out of $3 \ln (4)$ .

$x = \frac{3 \ln (0.00097656) + 3 \ln (4)}{18 \ln (0.125) + 5 \ln (4)}$

Step 9.3.3.2.3

Factor $3$ out of $3 \ln (0.00097656) + 3 \ln (4)$ .

$x = \frac{3 (\ln (0.00097656) + \ln (4))}{18 \ln (0.125) + 5 \ln (4)}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 (\ln (0.00097656) + \ln (4))}{18 \ln (0.125) + 5 \ln (4)}$

Decimal Form:

$x = 0.\overline{54}$