Algebra Examples

$1346 = 2870 {(0.99)}^{0.23 x + 1} - 1354$

Step 1

Rewrite the equation as $2870 \cdot {0.99}^{0.23 x + 1} - 1354 = 1346$ .

$2870 \cdot {0.99}^{0.23 x + 1} - 1354 = 1346$

Step 2

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

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Step 2.1

Add $1354$ to both sides of the equation.

$2870 \cdot {0.99}^{0.23 x + 1} = 1346 + 1354$

Step 2.2

Add $1346$ and $1354$ .

$2870 \cdot {0.99}^{0.23 x + 1} = 2700$

Step 3

Divide each term in $2870 \cdot {0.99}^{0.23 x + 1} = 2700$ by $2870$ and simplify.

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Step 3.1

Divide each term in $2870 \cdot {0.99}^{0.23 x + 1} = 2700$ by $2870$ .

$\frac{2870 \cdot {0.99}^{0.23 x + 1}}{2870} = \frac{2700}{2870}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2870$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2870 \cdot {0.99}^{0.23 x + 1}}{2870} = \frac{2700}{2870}$

Step 3.2.1.2

Divide ${0.99}^{0.23 x + 1}$ by $1$ .

${0.99}^{0.23 x + 1} = \frac{2700}{2870}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $2700$ and $2870$ .

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Step 3.3.1.1

Factor $10$ out of $2700$ .

${0.99}^{0.23 x + 1} = \frac{10 (270)}{2870}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $10$ out of $2870$ .

${0.99}^{0.23 x + 1} = \frac{10 \cdot 270}{10 \cdot 287}$

Step 3.3.1.2.2

Cancel the common factor.

${0.99}^{0.23 x + 1} = \frac{10 \cdot 270}{10 \cdot 287}$

Step 3.3.1.2.3

Rewrite the expression.

${0.99}^{0.23 x + 1} = \frac{270}{287}$

Step 4

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

$\ln ({0.99}^{0.23 x + 1}) = \ln (\frac{270}{287})$

Step 5

Expand $\ln ({0.99}^{0.23 x + 1})$ by moving $0.23 x + 1$ outside the logarithm.

$(0.23 x + 1) \ln (0.99) = \ln (\frac{270}{287})$

Step 6

Simplify the left side.

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Step 6.1

Simplify $(0.23 x + 1) \ln (0.99)$ .

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Step 6.1.1

Apply the distributive property.

$0.23 x \ln (0.99) + 1 \ln (0.99) = \ln (\frac{270}{287})$

Step 6.1.2

Multiply $\ln (0.99)$ by $0.23$ .

$- 0.00231157 x + 1 \ln (0.99) = \ln (\frac{270}{287})$

Step 6.1.3

Multiply $\ln (0.99)$ by $1$ .

$- 0.00231157 x + \ln (0.99) = \ln (\frac{270}{287})$

Step 7

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

$\ln (0.99) - \ln (\frac{270}{287}) = 0.00231157 x$

Step 8

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

$\ln (\frac{0.99}{\frac{270}{287}}) = 0.00231157 x$

Step 9

Multiply the numerator by the reciprocal of the denominator.

$\ln (0.99 (\frac{287}{270})) = 0.00231157 x$

Step 10

Multiply $0.99 (\frac{287}{270})$ .

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Step 10.1

Combine $0.99$ and $\frac{287}{270}$ .

$\ln (\frac{0.99 \cdot 287}{270}) = 0.00231157 x$

Step 10.2

Multiply $0.99$ by $287$ .

$\ln (\frac{284.13}{270}) = 0.00231157 x$

Step 11

Divide $284.13$ by $270$ .

$\ln (1.052\overline{3}) = 0.00231157 x$

Step 12

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$0.00231157 x = \ln (1.052\overline{3})$

Step 13

Divide each term in $0.00231157 x = \ln (1.052\overline{3})$ by $0.00231157$ and simplify.

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Step 13.1

Divide each term in $0.00231157 x = \ln (1.052\overline{3})$ by $0.00231157$ .

$\frac{0.00231157 x}{0.00231157} = \frac{\ln (1.052\overline{3})}{0.00231157}$

Step 13.2

Simplify the left side.

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Step 13.2.1

Cancel the common factor of $0.00231157$ .

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Step 13.2.1.1

Cancel the common factor.

$\frac{0.00231157 x}{0.00231157} = \frac{\ln (1.052\overline{3})}{0.00231157}$

Step 13.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (1.052\overline{3})}{0.00231157}$

Step 13.3

Simplify the right side.

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Step 13.3.1

Replace $e$ with an approximation.

$x = \frac{\log_{2.71828182} (1.052\overline{3})}{0.00231157}$

Step 13.3.2

Log base $2.71828182$ of $1.052\overline{3}$ is approximately $0.05100992$ .

$x = \frac{0.05100992}{0.00231157}$

Step 13.3.3

Divide $0.05100992$ by $0.00231157$ .

$x = 22.0671496$