Calculus Examples

$10^{- x} = 7^{6 x}$

Step 1

Take the log of both sides of the equation.

$\ln (10^{- x}) = \ln (7^{6 x})$

Step 2

Expand $\ln (10^{- x})$ by moving $- x$ outside the logarithm.

$- x \ln (10) = \ln (7^{6 x})$

Step 3

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

$- x \ln (10) = 6 x \ln (7)$

Step 4

Solve the equation for $x$ .

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Step 4.1

Subtract $6 x \ln (7)$ from both sides of the equation.

$- x \ln (10) - 6 x \ln (7) = 0$

Step 4.2

Factor $x$ out of $- x \ln (10) - 6 x \ln (7)$ .

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Step 4.2.1

Factor $x$ out of $- x \ln (10)$ .

$x (- 1 \ln (10)) - 6 x \ln (7) = 0$

Step 4.2.2

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

$x (- 1 \ln (10)) + x (- 6 \ln (7)) = 0$

Step 4.2.3

Factor $x$ out of $x (- 1 \ln (10)) + x (- 6 \ln (7))$ .

$x (- 1 \ln (10) - 6 \ln (7)) = 0$

Step 4.3

Rewrite $- 1 \ln (10)$ as $- \ln (10)$ .

$x (- \ln (10) - 6 \ln (7)) = 0$

Step 4.4

Divide each term in $x (- \ln (10) - 6 \ln (7)) = 0$ by $- \ln (10) - 6 \ln (7)$ and simplify.

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Step 4.4.1

Divide each term in $x (- \ln (10) - 6 \ln (7)) = 0$ by $- \ln (10) - 6 \ln (7)$ .

$\frac{x (- \ln (10) - 6 \ln (7))}{- \ln (10) - 6 \ln (7)} = \frac{0}{- \ln (10) - 6 \ln (7)}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $- \ln (10) - 6 \ln (7)$ .

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{x (- \ln (10) - 6 \ln (7))}{- \ln (10) - 6 \ln (7)} = \frac{0}{- \ln (10) - 6 \ln (7)}$

Step 4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{- \ln (10) - 6 \ln (7)}$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Divide $0$ by $- \ln (10) - 6 \ln (7)$ .

$x = 0$