Algebra Examples

$7^{x} = 6^{x + 7}$

Step 1

Take the log of both sides of the equation.

$\ln (7^{x}) = \ln (6^{x + 7})$

Step 2

Expand $\ln (7^{x})$ by moving $x$ outside the logarithm.

$x \ln (7) = \ln (6^{x + 7})$

Step 3

Expand $\ln (6^{x + 7})$ by moving $x + 7$ outside the logarithm.

$x \ln (7) = (x + 7) \ln (6)$

Step 4

Solve the equation for $x$ .

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Step 4.1

Apply the distributive property.

$x \ln (7) = x \ln (6) + 7 \ln (6)$

Step 4.2

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

$x \ln (7) - x \ln (6) = 7 \ln (6)$

Step 4.3

Factor $x$ out of $x \ln (7) - x \ln (6)$ .

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Step 4.3.1

Factor $x$ out of $x \ln (7)$ .

$x (\ln (7)) - x \ln (6) = 7 \ln (6)$

Step 4.3.2

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

$x (\ln (7)) + x (- 1 \ln (6)) = 7 \ln (6)$

Step 4.3.3

Factor $x$ out of $x (\ln (7)) + x (- 1 \ln (6))$ .

$x (\ln (7) - 1 \ln (6)) = 7 \ln (6)$

Step 4.4

Rewrite $- 1 \ln (6)$ as $- \ln (6)$ .

$x (\ln (7) - \ln (6)) = 7 \ln (6)$

Step 4.5

Divide each term in $x (\ln (7) - \ln (6)) = 7 \ln (6)$ by $\ln (7) - \ln (6)$ and simplify.

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Step 4.5.1

Divide each term in $x (\ln (7) - \ln (6)) = 7 \ln (6)$ by $\ln (7) - \ln (6)$ .

$\frac{x (\ln (7) - \ln (6))}{\ln (7) - \ln (6)} = \frac{7 \ln (6)}{\ln (7) - \ln (6)}$

Step 4.5.2

Simplify the left side.

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Step 4.5.2.1

Cancel the common factor of $\ln (7) - \ln (6)$ .

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Step 4.5.2.1.1

Cancel the common factor.

$\frac{x (\ln (7) - \ln (6))}{\ln (7) - \ln (6)} = \frac{7 \ln (6)}{\ln (7) - \ln (6)}$

Step 4.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{7 \ln (6)}{\ln (7) - \ln (6)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7 \ln (6)}{\ln (7) - \ln (6)}$

Decimal Form:

$x = 81.36400240 \dots$