Algebra Examples

${0.01}^{x + 1} = 1000^{x - 9}$

Step 1

Take the log of both sides of the equation.

$\ln ({0.01}^{x + 1}) = \ln (1000^{x - 9})$

Step 2

Expand $\ln ({0.01}^{x + 1})$ by moving $x + 1$ outside the logarithm.

$(x + 1) \ln (0.01) = \ln (1000^{x - 9})$

Step 3

Expand $\ln (1000^{x - 9})$ by moving $x - 9$ outside the logarithm.

$(x + 1) \ln (0.01) = (x - 9) \ln (1000)$

Step 4

Solve the equation for $x$ .

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Step 4.1

Simplify $(x + 1) \ln (0.01)$ .

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Step 4.1.1

Rewrite.

$0 + 0 + (x + 1) \ln (0.01) = (x - 9) \ln (1000)$

Step 4.1.2

Simplify by adding zeros.

$(x + 1) \ln (0.01) = (x - 9) \ln (1000)$

Step 4.1.3

Apply the distributive property.

$x \ln (0.01) + 1 \ln (0.01) = (x - 9) \ln (1000)$

Step 4.1.4

Multiply $\ln (0.01)$ by $1$ .

$x \ln (0.01) + \ln (0.01) = (x - 9) \ln (1000)$

Step 4.2

Apply the distributive property.

$x \ln (0.01) + \ln (0.01) = x \ln (1000) - 9 \ln (1000)$

Step 4.3

Subtract $x \ln (1000)$ from both sides of the equation.

$x \ln (0.01) + \ln (0.01) - x \ln (1000) = - 9 \ln (1000)$

Step 4.4

Subtract $\ln (0.01)$ from both sides of the equation.

$x \ln (0.01) - x \ln (1000) = - 9 \ln (1000) - \ln (0.01)$

Step 4.5

Factor $x$ out of $x \ln (0.01) - x \ln (1000)$ .

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Step 4.5.1

Factor $x$ out of $x \ln (0.01)$ .

$x (\ln (0.01)) - x \ln (1000) = - 9 \ln (1000) - \ln (0.01)$

Step 4.5.2

Factor $x$ out of $- x \ln (1000)$ .

$x (\ln (0.01)) + x (- 1 \ln (1000)) = - 9 \ln (1000) - \ln (0.01)$

Step 4.5.3

Factor $x$ out of $x (\ln (0.01)) + x (- 1 \ln (1000))$ .

$x (\ln (0.01) - 1 \ln (1000)) = - 9 \ln (1000) - \ln (0.01)$

Step 4.6

Rewrite $- 1 \ln (1000)$ as $- \ln (1000)$ .

$x (\ln (0.01) - \ln (1000)) = - 9 \ln (1000) - \ln (0.01)$

Step 4.7

Divide each term in $x (\ln (0.01) - \ln (1000)) = - 9 \ln (1000) - \ln (0.01)$ by $\ln (0.01) - \ln (1000)$ and simplify.

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Step 4.7.1

Divide each term in $x (\ln (0.01) - \ln (1000)) = - 9 \ln (1000) - \ln (0.01)$ by $\ln (0.01) - \ln (1000)$ .

$\frac{x (\ln (0.01) - \ln (1000))}{\ln (0.01) - \ln (1000)} = \frac{- 9 \ln (1000)}{\ln (0.01) - \ln (1000)} + \frac{- \ln (0.01)}{\ln (0.01) - \ln (1000)}$

Step 4.7.2

Simplify the left side.

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Step 4.7.2.1

Cancel the common factor of $\ln (0.01) - \ln (1000)$ .

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Step 4.7.2.1.1

Cancel the common factor.

$\frac{x (\ln (0.01) - \ln (1000))}{\ln (0.01) - \ln (1000)} = \frac{- 9 \ln (1000)}{\ln (0.01) - \ln (1000)} + \frac{- \ln (0.01)}{\ln (0.01) - \ln (1000)}$

Step 4.7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 9 \ln (1000)}{\ln (0.01) - \ln (1000)} + \frac{- \ln (0.01)}{\ln (0.01) - \ln (1000)}$

Step 4.7.3

Simplify the right side.

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Step 4.7.3.1

Combine the numerators over the common denominator.

$x = \frac{- 9 \ln (1000) - \ln (0.01)}{\ln (0.01) - \ln (1000)}$

Step 4.7.3.2

Factor $- 1$ out of $- 9 \ln (1000)$ .

$x = \frac{- (9 \ln (1000)) - \ln (0.01)}{\ln (0.01) - \ln (1000)}$

Step 4.7.3.3

Factor $- 1$ out of $- \ln (0.01)$ .

$x = \frac{- (9 \ln (1000)) - \ln (0.01)}{\ln (0.01) - \ln (1000)}$

Step 4.7.3.4

Factor $- 1$ out of $- (9 \ln (1000)) - \ln (0.01)$ .

$x = \frac{- (9 \ln (1000) + \ln (0.01))}{\ln (0.01) - \ln (1000)}$

Step 4.7.3.5

Simplify the expression.

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Step 4.7.3.5.1

Rewrite $- (9 \ln (1000) + \ln (0.01))$ as $- 1 (9 \ln (1000) + \ln (0.01))$ .

$x = \frac{- 1 (9 \ln (1000) + \ln (0.01))}{\ln (0.01) - \ln (1000)}$

Step 4.7.3.5.2

Move the negative in front of the fraction.

$x = - \frac{9 \ln (1000) + \ln (0.01)}{\ln (0.01) - \ln (1000)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{9 \ln (1000) + \ln (0.01)}{\ln (0.01) - \ln (1000)}$

Decimal Form:

$x = 5$