Precalculus Examples

$17^{6 x} = 18^{x + 2}$

Step 1

Take the log of both sides of the equation.

$\ln (17^{6 x}) = \ln (18^{x + 2})$

Step 2

Expand $\ln (17^{6 x})$ by moving $6 x$ outside the logarithm.

$6 x \ln (17) = \ln (18^{x + 2})$

Step 3

Expand $\ln (18^{x + 2})$ by moving $x + 2$ outside the logarithm.

$6 x \ln (17) = (x + 2) \ln (18)$

Step 4

Solve the equation for $x$ .

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Step 4.1

Apply the distributive property.

$6 x \ln (17) = x \ln (18) + 2 \ln (18)$

Step 4.2

Subtract $x \ln (18)$ from both sides of the equation.

$6 x \ln (17) - x \ln (18) = 2 \ln (18)$

Step 4.3

Factor $x$ out of $6 x \ln (17) - x \ln (18)$ .

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Step 4.3.1

Factor $x$ out of $6 x \ln (17)$ .

$x (6 \ln (17)) - x \ln (18) = 2 \ln (18)$

Step 4.3.2

Factor $x$ out of $- x \ln (18)$ .

$x (6 \ln (17)) + x (- 1 \ln (18)) = 2 \ln (18)$

Step 4.3.3

Factor $x$ out of $x (6 \ln (17)) + x (- 1 \ln (18))$ .

$x (6 \ln (17) - 1 \ln (18)) = 2 \ln (18)$

Step 4.4

Rewrite $- 1 \ln (18)$ as $- \ln (18)$ .

$x (6 \ln (17) - \ln (18)) = 2 \ln (18)$

Step 4.5

Divide each term in $x (6 \ln (17) - \ln (18)) = 2 \ln (18)$ by $6 \ln (17) - \ln (18)$ and simplify.

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Step 4.5.1

Divide each term in $x (6 \ln (17) - \ln (18)) = 2 \ln (18)$ by $6 \ln (17) - \ln (18)$ .

$\frac{x (6 \ln (17) - \ln (18))}{6 \ln (17) - \ln (18)} = \frac{2 \ln (18)}{6 \ln (17) - \ln (18)}$

Step 4.5.2

Simplify the left side.

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Step 4.5.2.1

Cancel the common factor of $6 \ln (17) - \ln (18)$ .

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Step 4.5.2.1.1

Cancel the common factor.

$\frac{x (6 \ln (17) - \ln (18))}{6 \ln (17) - \ln (18)} = \frac{2 \ln (18)}{6 \ln (17) - \ln (18)}$

Step 4.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{2 \ln (18)}{6 \ln (17) - \ln (18)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2 \ln (18)}{6 \ln (17) - \ln (18)}$

Decimal Form:

$x = 0.40972294 \dots$