Pre-Algebra Examples

$- {4.5}^{x + 1} = - 5$

Step 1

Divide each term in $- {4.5}^{x + 1} = - 5$ by $- 1$ and simplify.

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Step 1.1

Divide each term in $- {4.5}^{x + 1} = - 5$ by $- 1$ .

$\frac{- {4.5}^{x + 1}}{- 1} = \frac{- 5}{- 1}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Dividing two negative values results in a positive value.

$\frac{{4.5}^{x + 1}}{1} = \frac{- 5}{- 1}$

Step 1.2.2

Divide ${4.5}^{x + 1}$ by $1$ .

${4.5}^{x + 1} = \frac{- 5}{- 1}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $- 5$ by $- 1$ .

${4.5}^{x + 1} = 5$

Step 2

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

$\ln ({4.5}^{x + 1}) = \ln (5)$

Step 3

Expand $\ln ({4.5}^{x + 1})$ by moving $x + 1$ outside the logarithm.

$(x + 1) \ln (4.5) = \ln (5)$

Step 4

Simplify the left side.

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Step 4.1

Simplify $(x + 1) \ln (4.5)$ .

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Step 4.1.1

Apply the distributive property.

$x \ln (4.5) + 1 \ln (4.5) = \ln (5)$

Step 4.1.2

Multiply $\ln (4.5)$ by $1$ .

$x \ln (4.5) + \ln (4.5) = \ln (5)$

Step 5

Move all the terms containing a logarithm to the left side of the equation.

$x \ln (4.5) + \ln (4.5) - \ln (5) = 0$

Step 6

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$x \ln (4.5) + \ln (\frac{4.5}{5}) = 0$

Step 7

Divide $4.5$ by $5$ .

$x \ln (4.5) + \ln (0.9) = 0$

Step 8

Subtract $\ln (0.9)$ from both sides of the equation.

$x \ln (4.5) = - \ln (0.9)$

Step 9

Divide each term in $x \ln (4.5) = - \ln (0.9)$ by $\ln (4.5)$ and simplify.

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Step 9.1

Divide each term in $x \ln (4.5) = - \ln (0.9)$ by $\ln (4.5)$ .

$\frac{x \ln (4.5)}{\ln (4.5)} = \frac{- \ln (0.9)}{\ln (4.5)}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $\ln (4.5)$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{x \ln (4.5)}{\ln (4.5)} = \frac{- \ln (0.9)}{\ln (4.5)}$

Step 9.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \ln (0.9)}{\ln (4.5)}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (0.9)}{\ln (4.5)}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (0.9)}{\ln (4.5)}$

Decimal Form:

$x = 0.07004992 \dots$