Finite Math Examples

$9^{2 x + 1} = 128$

Step 1

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

$\ln (9^{2 x + 1}) = \ln (128)$

Step 2

Expand $\ln (9^{2 x + 1})$ by moving $2 x + 1$ outside the logarithm.

$(2 x + 1) \ln (9) = \ln (128)$

Step 3

Simplify the left side.

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Step 3.1

Simplify $(2 x + 1) \ln (9)$ .

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Step 3.1.1

Apply the distributive property.

$2 x \ln (9) + 1 \ln (9) = \ln (128)$

Step 3.1.2

Multiply $\ln (9)$ by $1$ .

$2 x \ln (9) + \ln (9) = \ln (128)$

Step 4

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

$2 x \ln (9) + \ln (9) - \ln (128) = 0$

Step 5

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$2 x \ln (9) + \ln (\frac{9}{128}) = 0$

Step 6

Subtract $\ln (\frac{9}{128})$ from both sides of the equation.

$2 x \ln (9) = - \ln (\frac{9}{128})$

Step 7

Divide each term in $2 x \ln (9) = - \ln (\frac{9}{128})$ by $2 \ln (9)$ and simplify.

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Step 7.1

Divide each term in $2 x \ln (9) = - \ln (\frac{9}{128})$ by $2 \ln (9)$ .

$\frac{2 x \ln (9)}{2 \ln (9)} = \frac{- \ln (\frac{9}{128})}{2 \ln (9)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $2$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{2 x \ln (9)}{2 \ln (9)} = \frac{- \ln (\frac{9}{128})}{2 \ln (9)}$

Step 7.2.1.2

Rewrite the expression.

$\frac{x \ln (9)}{\ln (9)} = \frac{- \ln (\frac{9}{128})}{2 \ln (9)}$

Step 7.2.2

Cancel the common factor of $\ln (9)$ .

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Step 7.2.2.1

Cancel the common factor.

$\frac{x \ln (9)}{\ln (9)} = \frac{- \ln (\frac{9}{128})}{2 \ln (9)}$

Step 7.2.2.2

Divide $x$ by $1$ .

$x = \frac{- \ln (\frac{9}{128})}{2 \ln (9)}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{9}{128})}{2 \ln (9)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (\frac{9}{128})}{2 \ln (9)}$

Decimal Form:

$x = 0.60412706 \dots$