Solve for x 3500=720(1.1)^(x/4)+1400

$3500 = 720 {(1.1)}^{\frac{x}{4}} + 1400$

Step 1

Rewrite the equation as $720 \cdot {1.1}^{\frac{x}{4}} + 1400 = 3500$ .

$720 \cdot {1.1}^{\frac{x}{4}} + 1400 = 3500$

Step 2

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

Subtract $1400$ from both sides of the equation.

$720 \cdot {1.1}^{\frac{x}{4}} = 3500 - 1400$

Subtract $1400$ from $3500$ .

$720 \cdot {1.1}^{\frac{x}{4}} = 2100$

Step 3

Divide each term in $(720 \cdot {1.1}^{\frac{x}{4}}) = 2100$ by $720$ and simplify.

Divide each term in $(720 \cdot {1.1}^{\frac{x}{4}}) = 2100$ by $720$ .

$\frac{(720 \cdot {1.1}^{\frac{x}{4}})}{720} = \frac{2100}{720}$

Simplify the left side.

Cancel the common factor of $720$ .

Cancel the common factor.

$\frac{720 \cdot {1.1}^{\frac{x}{4}}}{720} = \frac{2100}{720}$

Divide ${1.1}^{\frac{x}{4}}$ by $1$ .

${1.1}^{\frac{x}{4}} = \frac{2100}{720}$

Simplify the right side.

Cancel the common factor of $2100$ and $720$ .

Factor $60$ out of $2100$ .

${1.1}^{\frac{x}{4}} = \frac{60 (35)}{720}$

Cancel the common factors.

Factor $60$ out of $720$ .

${1.1}^{\frac{x}{4}} = \frac{60 \cdot 35}{60 \cdot 12}$

Cancel the common factor.

${1.1}^{\frac{x}{4}} = \frac{60 \cdot 35}{60 \cdot 12}$

Rewrite the expression.

${1.1}^{\frac{x}{4}} = \frac{35}{12}$

Step 4

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

$\ln ({1.1}^{\frac{x}{4}}) = \ln (\frac{35}{12})$

Step 5

Expand the left side.

Expand $\ln ({1.1}^{\frac{x}{4}})$ by moving $\frac{x}{4}$ outside the logarithm.

$\frac{x}{4} \ln (1.1) = \ln (\frac{35}{12})$

Combine $\frac{x}{4}$ and $\ln (1.1)$ .

$\frac{x \ln (1.1)}{4} = \ln (\frac{35}{12})$

Step 6

Multiply both sides of the equation by $\frac{4}{\ln (1.1)}$ .

$\frac{4}{\ln (1.1)} \cdot \frac{x \ln (1.1)}{4} = \frac{4}{\ln (1.1)} \ln (\frac{35}{12})$

Step 7

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{4}{\ln (1.1)} \cdot \frac{x \ln (1.1)}{4}$ .

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4}{\ln (1.1)} \cdot \frac{x \ln (1.1)}{4} = \frac{4}{\ln (1.1)} \ln (\frac{35}{12})$

Rewrite the expression.

$\frac{1}{\ln (1.1)} (x \ln (1.1)) = \frac{4}{\ln (1.1)} \ln (\frac{35}{12})$

Cancel the common factor of $\ln (1.1)$ .

Factor $\ln (1.1)$ out of $x \ln (1.1)$ .

$\frac{1}{\ln (1.1)} (\ln (1.1) x) = \frac{4}{\ln (1.1)} \ln (\frac{35}{12})$

Cancel the common factor.

$\frac{1}{\ln (1.1)} (\ln (1.1) x) = \frac{4}{\ln (1.1)} \ln (\frac{35}{12})$

Rewrite the expression.

$x = \frac{4}{\ln (1.1)} \ln (\frac{35}{12})$

Simplify the right side.

Combine $\frac{4}{\ln (1.1)}$ and $\ln (\frac{35}{12})$ .

$x = \frac{4 \ln (\frac{35}{12})}{\ln (1.1)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4 \ln (\frac{35}{12})}{\ln (1.1)}$

Decimal Form:

$x = 44.92453645 \dots$

Solve for x 3500=720(1.1)^(x/4)+1400

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