Calculus Examples

$3^{x - 5} = 7^{2 x}$

Step 1

Take the log of both sides of the equation.

$\ln (3^{x - 5}) = \ln (7^{2 x})$

Step 2

Expand $\ln (3^{x - 5})$ by moving $x - 5$ outside the logarithm.

$(x - 5) \ln (3) = \ln (7^{2 x})$

Step 3

Expand $\ln (7^{2 x})$ by moving $2 x$ outside the logarithm.

$(x - 5) \ln (3) = 2 x \ln (7)$

Step 4

Solve the equation for $x$ .

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Step 4.1

Simplify $(x - 5) \ln (3)$ .

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Step 4.1.1

Rewrite.

$0 + 0 + (x - 5) \ln (3) = 2 x \ln (7)$

Step 4.1.2

Simplify by adding zeros.

$(x - 5) \ln (3) = 2 x \ln (7)$

Step 4.1.3

Apply the distributive property.

$x \ln (3) - 5 \ln (3) = 2 x \ln (7)$

Step 4.2

Subtract $2 x \ln (7)$ from both sides of the equation.

$x \ln (3) - 5 \ln (3) - 2 x \ln (7) = 0$

Step 4.3

Add $5 \ln (3)$ to both sides of the equation.

$x \ln (3) - 2 x \ln (7) = 5 \ln (3)$

Step 4.4

Factor $x$ out of $x \ln (3) - 2 x \ln (7)$ .

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Step 4.4.1

Factor $x$ out of $x \ln (3)$ .

$x (\ln (3)) - 2 x \ln (7) = 5 \ln (3)$

Step 4.4.2

Factor $x$ out of $- 2 x \ln (7)$ .

$x (\ln (3)) + x (- 2 \ln (7)) = 5 \ln (3)$

Step 4.4.3

Factor $x$ out of $x (\ln (3)) + x (- 2 \ln (7))$ .

$x (\ln (3) - 2 \ln (7)) = 5 \ln (3)$

Step 4.5

Divide each term in $x (\ln (3) - 2 \ln (7)) = 5 \ln (3)$ by $\ln (3) - 2 \ln (7)$ and simplify.

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Step 4.5.1

Divide each term in $x (\ln (3) - 2 \ln (7)) = 5 \ln (3)$ by $\ln (3) - 2 \ln (7)$ .

$\frac{x (\ln (3) - 2 \ln (7))}{\ln (3) - 2 \ln (7)} = \frac{5 \ln (3)}{\ln (3) - 2 \ln (7)}$

Step 4.5.2

Simplify the left side.

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Step 4.5.2.1

Cancel the common factor of $\ln (3) - 2 \ln (7)$ .

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Step 4.5.2.1.1

Cancel the common factor.

$\frac{x (\ln (3) - 2 \ln (7))}{\ln (3) - 2 \ln (7)} = \frac{5 \ln (3)}{\ln (3) - 2 \ln (7)}$

Step 4.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{5 \ln (3)}{\ln (3) - 2 \ln (7)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \ln (3)}{\ln (3) - 2 \ln (7)}$

Decimal Form:

$x = - 1.96657800 \dots$