Solve for x 333=5230(0.95)^x-217

$333 = 5230 {(0.95)}^{x} - 217$

Step 1

Rewrite the equation as $5230 \cdot {0.95}^{x} - 217 = 333$ .

$5230 \cdot {0.95}^{x} - 217 = 333$

Step 2

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

Add $217$ to both sides of the equation.

$5230 \cdot {0.95}^{x} = 333 + 217$

Add $333$ and $217$ .

$5230 \cdot {0.95}^{x} = 550$

Step 3

Divide each term in $(5230 \cdot {0.95}^{x}) = 550$ by $5230$ and simplify.

Divide each term in $(5230 \cdot {0.95}^{x}) = 550$ by $5230$ .

$\frac{(5230 \cdot {0.95}^{x})}{5230} = \frac{550}{5230}$

Simplify the left side.

Cancel the common factor of $5230$ .

Cancel the common factor.

$\frac{5230 \cdot {0.95}^{x}}{5230} = \frac{550}{5230}$

Divide ${0.95}^{x}$ by $1$ .

${0.95}^{x} = \frac{550}{5230}$

Simplify the right side.

Cancel the common factor of $550$ and $5230$ .

Factor $10$ out of $550$ .

${0.95}^{x} = \frac{10 (55)}{5230}$

Cancel the common factors.

Factor $10$ out of $5230$ .

${0.95}^{x} = \frac{10 \cdot 55}{10 \cdot 523}$

Cancel the common factor.

${0.95}^{x} = \frac{10 \cdot 55}{10 \cdot 523}$

Rewrite the expression.

${0.95}^{x} = \frac{55}{523}$

Step 4

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

$\ln ({0.95}^{x}) = \ln (\frac{55}{523})$

Step 5

Expand $\ln ({0.95}^{x})$ by moving $x$ outside the logarithm.

$x \ln (0.95) = \ln (\frac{55}{523})$

Step 6

Divide each term in $x \ln (0.95) = \ln (\frac{55}{523})$ by $\ln (0.95)$ and simplify.

Divide each term in $x \ln (0.95) = \ln (\frac{55}{523})$ by $\ln (0.95)$ .

$\frac{x \ln (0.95)}{\ln (0.95)} = \frac{\ln (\frac{55}{523})}{\ln (0.95)}$

Simplify the left side.

Cancel the common factor of $\ln (0.95)$ .

Cancel the common factor.

$\frac{x \ln (0.95)}{\ln (0.95)} = \frac{\ln (\frac{55}{523})}{\ln (0.95)}$

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{55}{523})}{\ln (0.95)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (\frac{55}{523})}{\ln (0.95)}$

Decimal Form:

$x = 43.90921475 \dots$

Solve for x 333=5230(0.95)^x-217

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