Algebra Examples

${(\frac{625}{256})}^{5 x - 4} = 1$

Step 1

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

$\ln ({(\frac{625}{256})}^{5 x - 4}) = \ln (1)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln ({(\frac{625}{256})}^{5 x - 4})$ by moving $5 x - 4$ outside the logarithm.

$(5 x - 4) \ln (\frac{625}{256}) = \ln (1)$

Step 2.2

Rewrite $\ln (\frac{625}{256})$ as $\ln (625) - \ln (256)$ .

$(5 x - 4) (\ln (625) - \ln (256)) = \ln (1)$

Step 2.3

Rewrite $\ln (256)$ as $\ln (2^{8})$ .

$(5 x - 4) (\ln (625) - \ln (2^{8})) = \ln (1)$

Step 2.4

Expand $\ln (2^{8})$ by moving $8$ outside the logarithm.

$(5 x - 4) (\ln (625) - (8 \ln (2))) = \ln (1)$

Step 2.5

Multiply $8$ by $- 1$ .

$(5 x - 4) (\ln (625) - 8 \ln (2)) = \ln (1)$

Step 3

Simplify the left side.

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Step 3.1

Simplify $(5 x - 4) (\ln (625) - 8 \ln (2))$ .

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Step 3.1.1

Expand $(5 x - 4) (\ln (625) - 8 \ln (2))$ using the FOIL Method.

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Step 3.1.1.1

Apply the distributive property.

$5 x (\ln (625) - 8 \ln (2)) - 4 (\ln (625) - 8 \ln (2)) = \ln (1)$

Step 3.1.1.2

Apply the distributive property.

$5 x \ln (625) + 5 x (- 8 \ln (2)) - 4 (\ln (625) - 8 \ln (2)) = \ln (1)$

Step 3.1.1.3

Apply the distributive property.

$5 x \ln (625) + 5 x (- 8 \ln (2)) - 4 \ln (625) - 4 (- 8 \ln (2)) = \ln (1)$

Step 3.1.2

Simplify each term.

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Step 3.1.2.1

Multiply $- 8$ by $5$ .

$5 x \ln (625) - 40 x \ln (2) - 4 \ln (625) - 4 (- 8 \ln (2)) = \ln (1)$

Step 3.1.2.2

Multiply $- 8$ by $- 4$ .

$5 x \ln (625) - 40 x \ln (2) - 4 \ln (625) + 32 \ln (2) = \ln (1)$

Step 4

Simplify the right side.

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Step 4.1

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

$5 x \ln (625) - 40 x \ln (2) - 4 \ln (625) + 32 \ln (2) = 0$

Step 5

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

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Step 5.1

Add $4 \ln (625)$ to both sides of the equation.

$5 x \ln (625) - 40 x \ln (2) + 32 \ln (2) = 4 \ln (625)$

Step 5.2

Subtract $32 \ln (2)$ from both sides of the equation.

$5 x \ln (625) - 40 x \ln (2) = 4 \ln (625) - 32 \ln (2)$

Step 6

Factor $5 x$ out of $5 x \ln (625) - 40 x \ln (2)$ .

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Step 6.1

Factor $5 x$ out of $- 40 x \ln (2)$ .

$5 x \ln (625) + 5 x (- 8 \ln (2)) = 4 \ln (625) - 32 \ln (2)$

Step 6.2

Factor $5 x$ out of $5 x \ln (625) + 5 x (- 8 \ln (2))$ .

$5 x (\ln (625) - 8 \ln (2)) = 4 \ln (625) - 32 \ln (2)$

Step 7

Divide each term in $5 x (\ln (625) - 8 \ln (2)) = 4 \ln (625) - 32 \ln (2)$ by $5 (\ln (625) - 8 \ln (2))$ and simplify.

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Step 7.1

Divide each term in $5 x (\ln (625) - 8 \ln (2)) = 4 \ln (625) - 32 \ln (2)$ by $5 (\ln (625) - 8 \ln (2))$ .

$\frac{5 x (\ln (625) - 8 \ln (2))}{5 (\ln (625) - 8 \ln (2))} = \frac{4 \ln (625)}{5 (\ln (625) - 8 \ln (2))} + \frac{- 32 \ln (2)}{5 (\ln (625) - 8 \ln (2))}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $5$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{5 x (\ln (625) - 8 \ln (2))}{5 (\ln (625) - 8 \ln (2))} = \frac{4 \ln (625)}{5 (\ln (625) - 8 \ln (2))} + \frac{- 32 \ln (2)}{5 (\ln (625) - 8 \ln (2))}$

Step 7.2.1.2

Rewrite the expression.

$\frac{x (\ln (625) - 8 \ln (2))}{\ln (625) - 8 \ln (2)} = \frac{4 \ln (625)}{5 (\ln (625) - 8 \ln (2))} + \frac{- 32 \ln (2)}{5 (\ln (625) - 8 \ln (2))}$

Step 7.2.2

Cancel the common factor of $\ln (625) - 8 \ln (2)$ .

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Step 7.2.2.1

Cancel the common factor.

$\frac{x (\ln (625) - 8 \ln (2))}{\ln (625) - 8 \ln (2)} = \frac{4 \ln (625)}{5 (\ln (625) - 8 \ln (2))} + \frac{- 32 \ln (2)}{5 (\ln (625) - 8 \ln (2))}$

Step 7.2.2.2

Divide $x$ by $1$ .

$x = \frac{4 \ln (625)}{5 (\ln (625) - 8 \ln (2))} + \frac{- 32 \ln (2)}{5 (\ln (625) - 8 \ln (2))}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Move the negative in front of the fraction.

$x = \frac{4 \ln (625)}{5 (\ln (625) - 8 \ln (2))} - \frac{32 \ln (2)}{5 (\ln (625) - 8 \ln (2))}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4 \ln (625)}{5 (\ln (625) - 8 \ln (2))} - \frac{32 \ln (2)}{5 (\ln (625) - 8 \ln (2))}$

Decimal Form:

$x = 0.8$