Solve for x 1400=870(1.01)^(x+3)

$1400 = 870 {(1.01)}^{x + 3}$

Step 1

Rewrite the equation as $870 \cdot {1.01}^{x + 3} = 1400$ .

$870 \cdot {1.01}^{x + 3} = 1400$

Step 2

Divide each term in $(870 \cdot {1.01}^{x + 3}) = 1400$ by $870$ and simplify.

Divide each term in $(870 \cdot {1.01}^{x + 3}) = 1400$ by $870$ .

$\frac{(870 \cdot {1.01}^{x + 3})}{870} = \frac{1400}{870}$

Simplify the left side.

Cancel the common factor of $870$ .

Cancel the common factor.

$\frac{870 \cdot {1.01}^{x + 3}}{870} = \frac{1400}{870}$

Divide ${1.01}^{x + 3}$ by $1$ .

${1.01}^{x + 3} = \frac{1400}{870}$

Simplify the right side.

Cancel the common factor of $1400$ and $870$ .

Factor $10$ out of $1400$ .

${1.01}^{x + 3} = \frac{10 (140)}{870}$

Cancel the common factors.

Factor $10$ out of $870$ .

${1.01}^{x + 3} = \frac{10 \cdot 140}{10 \cdot 87}$

Cancel the common factor.

${1.01}^{x + 3} = \frac{10 \cdot 140}{10 \cdot 87}$

Rewrite the expression.

${1.01}^{x + 3} = \frac{140}{87}$

Step 3

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

$\ln ({1.01}^{x + 3}) = \ln (\frac{140}{87})$

Step 4

Expand $\ln ({1.01}^{x + 3})$ by moving $x + 3$ outside the logarithm.

$(x + 3) \ln (1.01) = \ln (\frac{140}{87})$

Step 5

Simplify the left side.

Apply the distributive property.

$x \ln (1.01) + 3 \ln (1.01) = \ln (\frac{140}{87})$

Step 6

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

$x \ln (1.01) + 3 \ln (1.01) - \ln (\frac{140}{87}) = 0$

Step 7

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

Subtract $3 \ln (1.01)$ from both sides of the equation.

$x \ln (1.01) - \ln (\frac{140}{87}) = - 3 \ln (1.01)$

Add $\ln (\frac{140}{87})$ to both sides of the equation.

$x \ln (1.01) = - 3 \ln (1.01) + \ln (\frac{140}{87})$

Step 8

Divide each term in $x \ln (1.01) = - 3 \ln (1.01) + \ln (\frac{140}{87})$ by $\ln (1.01)$ and simplify.

Divide each term in $x \ln (1.01) = - 3 \ln (1.01) + \ln (\frac{140}{87})$ by $\ln (1.01)$ .

$\frac{x \ln (1.01)}{\ln (1.01)} = \frac{- 3 \ln (1.01)}{\ln (1.01)} + \frac{\ln (\frac{140}{87})}{\ln (1.01)}$

Simplify the left side.

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

Cancel the common factor.

$\frac{x \ln (1.01)}{\ln (1.01)} = \frac{- 3 \ln (1.01)}{\ln (1.01)} + \frac{\ln (\frac{140}{87})}{\ln (1.01)}$

Divide $x$ by $1$ .

$x = \frac{- 3 \ln (1.01)}{\ln (1.01)} + \frac{\ln (\frac{140}{87})}{\ln (1.01)}$

Simplify the right side.

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

Cancel the common factor.

$x = \frac{- 3 \ln (1.01)}{\ln (1.01)} + \frac{\ln (\frac{140}{87})}{\ln (1.01)}$

Divide $- 3$ by $1$ .

$x = - 3 + \frac{\ln (\frac{140}{87})}{\ln (1.01)}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - 3 + \frac{\ln (\frac{140}{87})}{\ln (1.01)}$

Decimal Form:

$x = 44.81090307 \dots$

Solve for x 1400=870(1.01)^(x+3)

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