Precalculus Examples

$6^{1 - 9 x} = 5^{x}$

Step 1

Take the log of both sides of the equation.

$\ln (6^{1 - 9 x}) = \ln (5^{x})$

Step 2

Expand $\ln (6^{1 - 9 x})$ by moving $1 - 9 x$ outside the logarithm.

$(1 - 9 x) \ln (6) = \ln (5^{x})$

Step 3

Expand $\ln (5^{x})$ by moving $x$ outside the logarithm.

$(1 - 9 x) \ln (6) = x \ln (5)$

Step 4

Solve the equation for $x$ .

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Step 4.1

Simplify $(1 - 9 x) \ln (6)$ .

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Step 4.1.1

Rewrite.

$0 + 0 + (1 - 9 x) \ln (6) = x \ln (5)$

Step 4.1.2

Simplify by adding zeros.

$(1 - 9 x) \ln (6) = x \ln (5)$

Step 4.1.3

Apply the distributive property.

$1 \ln (6) - 9 x \ln (6) = x \ln (5)$

Step 4.1.4

Multiply $\ln (6)$ by $1$ .

$\ln (6) - 9 x \ln (6) = x \ln (5)$

Step 4.2

Subtract $x \ln (5)$ from both sides of the equation.

$\ln (6) - 9 x \ln (6) - x \ln (5) = 0$

Step 4.3

Subtract $\ln (6)$ from both sides of the equation.

$- 9 x \ln (6) - x \ln (5) = - \ln (6)$

Step 4.4

Factor $x$ out of $- 9 x \ln (6) - x \ln (5)$ .

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Step 4.4.1

Factor $x$ out of $- 9 x \ln (6)$ .

$x (- 9 \ln (6)) - x \ln (5) = - \ln (6)$

Step 4.4.2

Factor $x$ out of $- x \ln (5)$ .

$x (- 9 \ln (6)) + x (- 1 \ln (5)) = - \ln (6)$

Step 4.4.3

Factor $x$ out of $x (- 9 \ln (6)) + x (- 1 \ln (5))$ .

$x (- 9 \ln (6) - 1 \ln (5)) = - \ln (6)$

Step 4.5

Rewrite $- 1 \ln (5)$ as $- \ln (5)$ .

$x (- 9 \ln (6) - \ln (5)) = - \ln (6)$

Step 4.6

Divide each term in $x (- 9 \ln (6) - \ln (5)) = - \ln (6)$ by $- 9 \ln (6) - \ln (5)$ and simplify.

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Step 4.6.1

Divide each term in $x (- 9 \ln (6) - \ln (5)) = - \ln (6)$ by $- 9 \ln (6) - \ln (5)$ .

$\frac{x (- 9 \ln (6) - \ln (5))}{- 9 \ln (6) - \ln (5)} = \frac{- \ln (6)}{- 9 \ln (6) - \ln (5)}$

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Cancel the common factor of $- 9 \ln (6) - \ln (5)$ .

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Step 4.6.2.1.1

Cancel the common factor.

$\frac{x (- 9 \ln (6) - \ln (5))}{- 9 \ln (6) - \ln (5)} = \frac{- \ln (6)}{- 9 \ln (6) - \ln (5)}$

Step 4.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \ln (6)}{- 9 \ln (6) - \ln (5)}$

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (6)}{- 9 \ln (6) - \ln (5)}$

Step 4.6.3.2

Factor $- 1$ out of $- 9 \ln (6)$ .

$x = - \frac{\ln (6)}{- (9 \ln (6)) - \ln (5)}$

Step 4.6.3.3

Factor $- 1$ out of $- \ln (5)$ .

$x = - \frac{\ln (6)}{- (9 \ln (6)) - \ln (5)}$

Step 4.6.3.4

Factor $- 1$ out of $- (9 \ln (6)) - \ln (5)$ .

$x = - \frac{\ln (6)}{- (9 \ln (6) + \ln (5))}$

Step 4.6.3.5

Simplify the expression.

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Step 4.6.3.5.1

Rewrite $- (9 \ln (6) + \ln (5))$ as $- 1 (9 \ln (6) + \ln (5))$ .

$x = - \frac{\ln (6)}{- 1 (9 \ln (6) + \ln (5))}$

Step 4.6.3.5.2

Move the negative in front of the fraction.

$x = - - \frac{\ln (6)}{9 \ln (6) + \ln (5)}$

Step 4.6.3.5.3

Multiply $- 1$ by $- 1$ .

$x = 1 \frac{\ln (6)}{9 \ln (6) + \ln (5)}$

Step 4.6.3.5.4

Multiply $\frac{\ln (6)}{9 \ln (6) + \ln (5)}$ by $1$ .

$x = \frac{\ln (6)}{9 \ln (6) + \ln (5)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (6)}{9 \ln (6) + \ln (5)}$

Decimal Form:

$x = 0.10102801 \dots$