Algebra Examples

$2.1 = {1.01}^{\frac{x}{5} - 4} + 1$

Step 1

Rewrite the equation as ${1.01}^{\frac{x}{5} - 4} + 1 = 2.1$ .

${1.01}^{\frac{x}{5} - 4} + 1 = 2.1$

Step 2

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

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Step 2.1

Subtract $1$ from both sides of the equation.

${1.01}^{\frac{x}{5} - 4} = 2.1 - 1$

Step 2.2

Subtract $1$ from $2.1$ .

${1.01}^{\frac{x}{5} - 4} = 1.1$

Step 3

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

$\ln ({1.01}^{\frac{x}{5} - 4}) = \ln (1.1)$

Step 4

Expand $\ln ({1.01}^{\frac{x}{5} - 4})$ by moving $\frac{x}{5} - 4$ outside the logarithm.

$(\frac{x}{5} - 4) \ln (1.01) = \ln (1.1)$

Step 5

Simplify the left side.

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Step 5.1

Simplify $(\frac{x}{5} - 4) \ln (1.01)$ .

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Step 5.1.1

Apply the distributive property.

$\frac{x}{5} \ln (1.01) - 4 \ln (1.01) = \ln (1.1)$

Step 5.1.2

Combine $\frac{x}{5}$ and $\ln (1.01)$ .

$\frac{x \ln (1.01)}{5} - 4 \ln (1.01) = \ln (1.1)$

Step 6

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

$\frac{x \ln (1.01)}{5} - 4 \ln (1.01) - \ln (1.1) = 0$

Step 7

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

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Step 7.1

Add $4 \ln (1.01)$ to both sides of the equation.

$\frac{x \ln (1.01)}{5} - \ln (1.1) = 4 \ln (1.01)$

Step 7.2

Add $\ln (1.1)$ to both sides of the equation.

$\frac{x \ln (1.01)}{5} = 4 \ln (1.01) + \ln (1.1)$

Step 8

Multiply both sides of the equation by $5$ .

$5 \frac{x \ln (1.01)}{5} = 5 (4 \ln (1.01) + \ln (1.1))$

Step 9

Simplify both sides of the equation.

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Step 9.1

Simplify the left side.

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Step 9.1.1

Cancel the common factor of $5$ .

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Step 9.1.1.1

Cancel the common factor.

$5 \frac{x \ln (1.01)}{5} = 5 (4 \ln (1.01) + \ln (1.1))$

Step 9.1.1.2

Rewrite the expression.

$x \ln (1.01) = 5 (4 \ln (1.01) + \ln (1.1))$

Step 9.2

Simplify the right side.

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Step 9.2.1

Simplify $5 (4 \ln (1.01) + \ln (1.1))$ .

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Step 9.2.1.1

Apply the distributive property.

$x \ln (1.01) = 5 (4 \ln (1.01)) + 5 \ln (1.1)$

Step 9.2.1.2

Multiply $4$ by $5$ .

$x \ln (1.01) = 20 \ln (1.01) + 5 \ln (1.1)$

Step 10

Divide each term in $x \ln (1.01) = 20 \ln (1.01) + 5 \ln (1.1)$ by $\ln (1.01)$ and simplify.

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Step 10.1

Divide each term in $x \ln (1.01) = 20 \ln (1.01) + 5 \ln (1.1)$ by $\ln (1.01)$ .

$\frac{x \ln (1.01)}{\ln (1.01)} = \frac{20 \ln (1.01)}{\ln (1.01)} + \frac{5 \ln (1.1)}{\ln (1.01)}$

Step 10.2

Simplify the left side.

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Step 10.2.1

Cancel the common factor of $\ln (1.01)$ .

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Step 10.2.1.1

Cancel the common factor.

$\frac{x \ln (1.01)}{\ln (1.01)} = \frac{20 \ln (1.01)}{\ln (1.01)} + \frac{5 \ln (1.1)}{\ln (1.01)}$

Step 10.2.1.2

Divide $x$ by $1$ .

$x = \frac{20 \ln (1.01)}{\ln (1.01)} + \frac{5 \ln (1.1)}{\ln (1.01)}$

Step 10.3

Simplify the right side.

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Step 10.3.1

Cancel the common factor of $\ln (1.01)$ .

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Step 10.3.1.1

Cancel the common factor.

$x = \frac{20 \ln (1.01)}{\ln (1.01)} + \frac{5 \ln (1.1)}{\ln (1.01)}$

Step 10.3.1.2

Divide $20$ by $1$ .

$x = 20 + \frac{5 \ln (1.1)}{\ln (1.01)}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = 20 + \frac{5 \ln (1.1)}{\ln (1.01)}$

Decimal Form:

$x = 67.89297019 \dots$