Basic Math Examples

$100 = 9000^{n - 1}$

Step 1

Rewrite the equation as $9000^{n - 1} = 100$ .

$9000^{n - 1} = 100$

Step 2

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

$\ln (9000^{n - 1}) = \ln (100)$

Step 3

Expand $\ln (9000^{n - 1})$ by moving $n - 1$ outside the logarithm.

$(n - 1) \ln (9000) = \ln (100)$

Step 4

Simplify the left side.

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Step 4.1

Simplify $(n - 1) \ln (9000)$ .

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Step 4.1.1

Apply the distributive property.

$n \ln (9000) - 1 \ln (9000) = \ln (100)$

Step 4.1.2

Rewrite $- 1 \ln (9000)$ as $- \ln (9000)$ .

$n \ln (9000) - \ln (9000) = \ln (100)$

Step 5

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

$n \ln (9000) - \ln (9000) - \ln (100) = 0$

Step 6

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

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Step 6.1

Add $\ln (9000)$ to both sides of the equation.

$n \ln (9000) - \ln (100) = \ln (9000)$

Step 6.2

Add $\ln (100)$ to both sides of the equation.

$n \ln (9000) = \ln (9000) + \ln (100)$

Step 7

Divide each term in $n \ln (9000) = \ln (9000) + \ln (100)$ by $\ln (9000)$ and simplify.

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Step 7.1

Divide each term in $n \ln (9000) = \ln (9000) + \ln (100)$ by $\ln (9000)$ .

$\frac{n \ln (9000)}{\ln (9000)} = \frac{\ln (9000)}{\ln (9000)} + \frac{\ln (100)}{\ln (9000)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $\ln (9000)$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{n \ln (9000)}{\ln (9000)} = \frac{\ln (9000)}{\ln (9000)} + \frac{\ln (100)}{\ln (9000)}$

Step 7.2.1.2

Divide $n$ by $1$ .

$n = \frac{\ln (9000)}{\ln (9000)} + \frac{\ln (100)}{\ln (9000)}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Cancel the common factor of $\ln (9000)$ .

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Step 7.3.1.1

Cancel the common factor.

$n = \frac{\ln (9000)}{\ln (9000)} + \frac{\ln (100)}{\ln (9000)}$

Step 7.3.1.2

Rewrite the expression.

$n = 1 + \frac{\ln (100)}{\ln (9000)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$n = 1 + \frac{\ln (100)}{\ln (9000)}$

Decimal Form:

$n = 1.50578587 \dots$