Solve for x 18(1.2)^(2x-1)=900

$18 {(1.2)}^{2 x - 1} = 900$

Step 1

Divide each term in $(18 \cdot {1.2}^{2 x - 1}) = 900$ by $18$ and simplify.

Divide each term in $(18 \cdot {1.2}^{2 x - 1}) = 900$ by $18$ .

$\frac{(18 \cdot {1.2}^{2 x - 1})}{18} = \frac{900}{18}$

Simplify the left side.

Cancel the common factor of $18$ .

Cancel the common factor.

$\frac{18 \cdot {1.2}^{2 x - 1}}{18} = \frac{900}{18}$

Divide ${1.2}^{2 x - 1}$ by $1$ .

${1.2}^{2 x - 1} = \frac{900}{18}$

Simplify the right side.

Divide $900$ by $18$ .

${1.2}^{2 x - 1} = 50$

Step 2

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

$\ln ({1.2}^{2 x - 1}) = \ln (50)$

Step 3

Expand $\ln ({1.2}^{2 x - 1})$ by moving $2 x - 1$ outside the logarithm.

$(2 x - 1) \ln (1.2) = \ln (50)$

Step 4

Simplify the left side.

Simplify $(2 x - 1) \ln (1.2)$ .

Apply the distributive property.

$2 x \ln (1.2) - 1 \ln (1.2) = \ln (50)$

Rewrite $- 1 \ln (1.2)$ as $- \ln (1.2)$ .

$2 x \ln (1.2) - \ln (1.2) = \ln (50)$

Step 5

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

$2 x \ln (1.2) - \ln (1.2) - \ln (50) = 0$

Step 6

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

Add $\ln (1.2)$ to both sides of the equation.

$2 x \ln (1.2) - \ln (50) = \ln (1.2)$

Add $\ln (50)$ to both sides of the equation.

$2 x \ln (1.2) = \ln (1.2) + \ln (50)$

Step 7

Divide each term in $2 x \ln (1.2) = \ln (1.2) + \ln (50)$ by $2 \ln (1.2)$ and simplify.

Divide each term in $2 x \ln (1.2) = \ln (1.2) + \ln (50)$ by $2 \ln (1.2)$ .

$\frac{2 x \ln (1.2)}{2 \ln (1.2)} = \frac{\ln (1.2)}{2 \ln (1.2)} + \frac{\ln (50)}{2 \ln (1.2)}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x \ln (1.2)}{2 \ln (1.2)} = \frac{\ln (1.2)}{2 \ln (1.2)} + \frac{\ln (50)}{2 \ln (1.2)}$

Rewrite the expression.

$\frac{x \ln (1.2)}{\ln (1.2)} = \frac{\ln (1.2)}{2 \ln (1.2)} + \frac{\ln (50)}{2 \ln (1.2)}$

Cancel the common factor of $\ln (1.2)$ .

Cancel the common factor.

$\frac{x \ln (1.2)}{\ln (1.2)} = \frac{\ln (1.2)}{2 \ln (1.2)} + \frac{\ln (50)}{2 \ln (1.2)}$

Divide $x$ by $1$ .

$x = \frac{\ln (1.2)}{2 \ln (1.2)} + \frac{\ln (50)}{2 \ln (1.2)}$

Simplify the right side.

Cancel the common factor of $\ln (1.2)$ .

Cancel the common factor.

$x = \frac{\ln (1.2)}{2 \ln (1.2)} + \frac{\ln (50)}{2 \ln (1.2)}$

Rewrite the expression.

$x = \frac{1}{2} + \frac{\ln (50)}{2 \ln (1.2)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{2} + \frac{\ln (50)}{2 \ln (1.2)}$

Decimal Form:

$x = 11.22836112 \dots$

Solve for x 18(1.2)^(2x-1)=900

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