Solve for x 1600=780(1.015)^(x/5)

$1600 = 780 {(1.015)}^{\frac{x}{5}}$

Rewrite the equation as $780 \cdot {1.015}^{\frac{x}{5}} = 1600$ .

$780 \cdot {1.015}^{\frac{x}{5}} = 1600$

Divide each term in $(780 \cdot {1.015}^{\frac{x}{5}}) = 1600$ by $780$ and simplify.

Divide each term in $(780 \cdot {1.015}^{\frac{x}{5}}) = 1600$ by $780$ .

$\frac{(780 \cdot {1.015}^{\frac{x}{5}})}{780} = \frac{1600}{780}$

Simplify the left side.

Cancel the common factor of $780$ .

Cancel the common factor.

$\frac{780 \cdot {1.015}^{\frac{x}{5}}}{780} = \frac{1600}{780}$

Divide ${1.015}^{\frac{x}{5}}$ by $1$ .

${1.015}^{\frac{x}{5}} = \frac{1600}{780}$

Simplify the right side.

Cancel the common factor of $1600$ and $780$ .

Factor $20$ out of $1600$ .

${1.015}^{\frac{x}{5}} = \frac{20 (80)}{780}$

Cancel the common factors.

Factor $20$ out of $780$ .

${1.015}^{\frac{x}{5}} = \frac{20 \cdot 80}{20 \cdot 39}$

Cancel the common factor.

${1.015}^{\frac{x}{5}} = \frac{20 \cdot 80}{20 \cdot 39}$

Rewrite the expression.

${1.015}^{\frac{x}{5}} = \frac{80}{39}$

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

$\ln ({1.015}^{\frac{x}{5}}) = \ln (\frac{80}{39})$

Expand the left side.

Expand $\ln ({1.015}^{\frac{x}{5}})$ by moving $\frac{x}{5}$ outside the logarithm.

$\frac{x}{5} \ln (1.015) = \ln (\frac{80}{39})$

Combine $\frac{x}{5}$ and $\ln (1.015)$ .

$\frac{x \ln (1.015)}{5} = \ln (\frac{80}{39})$

Multiply both sides of the equation by $\frac{5}{\ln (1.015)}$ .

$\frac{5}{\ln (1.015)} \cdot \frac{x \ln (1.015)}{5} = \frac{5}{\ln (1.015)} \ln (\frac{80}{39})$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{5}{\ln (1.015)} \cdot \frac{x \ln (1.015)}{5}$ .

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5}{\ln (1.015)} \cdot \frac{x \ln (1.015)}{5} = \frac{5}{\ln (1.015)} \ln (\frac{80}{39})$

Rewrite the expression.

$\frac{1}{\ln (1.015)} (x \ln (1.015)) = \frac{5}{\ln (1.015)} \ln (\frac{80}{39})$

Cancel the common factor of $\ln (1.015)$ .

Factor $\ln (1.015)$ out of $x \ln (1.015)$ .

$\frac{1}{\ln (1.015)} (\ln (1.015) x) = \frac{5}{\ln (1.015)} \ln (\frac{80}{39})$

Cancel the common factor.

$\frac{1}{\ln (1.015)} (\ln (1.015) x) = \frac{5}{\ln (1.015)} \ln (\frac{80}{39})$

Rewrite the expression.

$x = \frac{5}{\ln (1.015)} \ln (\frac{80}{39})$

Simplify the right side.

Combine $\frac{5}{\ln (1.015)}$ and $\ln (\frac{80}{39})$ .

$x = \frac{5 \ln (\frac{80}{39})}{\ln (1.015)}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \ln (\frac{80}{39})}{\ln (1.015)}$

Decimal Form:

$x = 241.28003494 \dots$

Solve for x 1600=780(1.015)^(x/5)

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