Trigonometry Examples

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

Step 1

Take the log of both sides of the equation.

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

Step 2

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

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

Step 3

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

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

Step 4

Solve the equation for $x$ .

Tap for more steps...

Step 4.1

Simplify $(x - 2) \ln (7)$ .

Tap for more steps...

Step 4.1.1

Rewrite.

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

Step 4.1.2

Simplify by adding zeros.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

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

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

Step 4.3

Add $2 \ln (7)$ to both sides of the equation.

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

Step 4.4

Factor $x$ out of $x \ln (7) - x \ln (5)$ .

Tap for more steps...

Step 4.4.1

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

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

Step 4.4.2

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

$x (\ln (7)) + x (- 1 \ln (5)) = 2 \ln (7)$

Step 4.4.3

Factor $x$ out of $x (\ln (7)) + x (- 1 \ln (5))$ .

$x (\ln (7) - 1 \ln (5)) = 2 \ln (7)$

Step 4.5

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

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

Step 4.6

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

Tap for more steps...

Step 4.6.1

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

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

Step 4.6.2

Simplify the left side.

Tap for more steps...

Step 4.6.2.1

Cancel the common factor of $\ln (7) - \ln (5)$ .

Tap for more steps...

Step 4.6.2.1.1

Cancel the common factor.

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

Step 4.6.2.1.2

Divide $x$ by $1$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 11.56654212 \dots$