Calculus Examples

$15 \cdot 3^{2 x} = 225$

Step 1

Divide each term in $15 \cdot 3^{2 x} = 225$ by $15$ and simplify.

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Step 1.1

Divide each term in $15 \cdot 3^{2 x} = 225$ by $15$ .

$\frac{15 \cdot 3^{2 x}}{15} = \frac{225}{15}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $15$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{15 \cdot 3^{2 x}}{15} = \frac{225}{15}$

Step 1.2.1.2

Divide $3^{2 x}$ by $1$ .

$3^{2 x} = \frac{225}{15}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $225$ by $15$ .

$3^{2 x} = 15$

Step 2

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

$\ln (3^{2 x}) = \ln (15)$

Step 3

Expand $\ln (3^{2 x})$ by moving $2 x$ outside the logarithm.

$2 x \ln (3) = \ln (15)$

Step 4

Divide each term in $2 x \ln (3) = \ln (15)$ by $2 \ln (3)$ and simplify.

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Step 4.1

Divide each term in $2 x \ln (3) = \ln (15)$ by $2 \ln (3)$ .

$\frac{2 x \ln (3)}{2 \ln (3)} = \frac{\ln (15)}{2 \ln (3)}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 x \ln (3)}{2 \ln (3)} = \frac{\ln (15)}{2 \ln (3)}$

Step 4.2.1.2

Rewrite the expression.

$\frac{x \ln (3)}{\ln (3)} = \frac{\ln (15)}{2 \ln (3)}$

Step 4.2.2

Cancel the common factor of $\ln (3)$ .

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Step 4.2.2.1

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{\ln (15)}{2 \ln (3)}$

Step 4.2.2.2

Divide $x$ by $1$ .

$x = \frac{\ln (15)}{2 \ln (3)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (15)}{2 \ln (3)}$

Decimal Form:

$x = 1.23248676 \dots$