Basic Math Examples

$27 \cdot 5^{3} = 15^{n}$

Step 1

Rewrite the equation as $15^{n} = 27 \cdot 5^{3}$ .

$15^{n} = 27 \cdot 5^{3}$

Step 2

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

$\ln (15^{n}) = \ln (27 \cdot 5^{3})$

Step 3

Expand $\ln (15^{n})$ by moving $n$ outside the logarithm.

$n \ln (15) = \ln (27 \cdot 5^{3})$

Step 4

Simplify the right side.

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Step 4.1

Simplify $\ln (27 \cdot 5^{3})$ .

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Step 4.1.1

Raise $5$ to the power of $3$ .

$n \ln (15) = \ln (27 \cdot 125)$

Step 4.1.2

Multiply $27$ by $125$ .

$n \ln (15) = \ln (3375)$

Step 5

Divide each term in $n \ln (15) = \ln (3375)$ by $\ln (15)$ and simplify.

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Step 5.1

Divide each term in $n \ln (15) = \ln (3375)$ by $\ln (15)$ .

$\frac{n \ln (15)}{\ln (15)} = \frac{\ln (3375)}{\ln (15)}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $\ln (15)$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{n \ln (15)}{\ln (15)} = \frac{\ln (3375)}{\ln (15)}$

Step 5.2.1.2

Divide $n$ by $1$ .

$n = \frac{\ln (3375)}{\ln (15)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$n = \frac{\ln (3375)}{\ln (15)}$

Decimal Form:

$n = 3$