Solve for x 7^(3x-1)=5^(x-1)

$7^{3 x - 1} = 5^{x - 1}$

Step 1

Take the log of both sides of the equation.

$\ln (7^{3 x - 1}) = \ln (5^{x - 1})$

Step 2

Expand $\ln (7^{3 x - 1})$ by moving $3 x - 1$ outside the logarithm.

$(3 x - 1) \ln (7) = \ln (5^{x - 1})$

Step 3

Expand $\ln (5^{x - 1})$ by moving $x - 1$ outside the logarithm.

$(3 x - 1) \ln (7) = (x - 1) \ln (5)$

Step 4

Solve the equation for $x$ .

Simplify $(3 x - 1) \ln (7)$ .

Rewrite.

$0 + 0 + (3 x - 1) \ln (7) = (x - 1) \ln (5)$

Simplify by adding zeros.

$(3 x - 1) \ln (7) = (x - 1) \ln (5)$

Apply the distributive property.

$3 x \ln (7) - 1 \ln (7) = (x - 1) \ln (5)$

Rewrite $- 1 \ln (7)$ as $- \ln (7)$ .

$3 x \ln (7) - \ln (7) = (x - 1) \ln (5)$

Simplify $(x - 1) \ln (5)$ .

Apply the distributive property.

$3 x \ln (7) - \ln (7) = x \ln (5) - 1 \ln (5)$

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

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

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

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

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

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

Factor $x$ out of $3 x \ln (7) - x \ln (5)$ .

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

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

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

$x (3 \ln (7)) + x (- 1 \ln (5)) = - \ln (5) + \ln (7)$

Factor $x$ out of $x (3 \ln (7)) + x (- 1 \ln (5))$ .

$x (3 \ln (7) - 1 \ln (5)) = - \ln (5) + \ln (7)$

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

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

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

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

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

Simplify the left side.

Cancel the common factor of $3 \ln (7) - \ln (5)$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Combine the numerators over the common denominator.

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

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

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

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

$x = \frac{- \ln (5) - 1 (- \ln (7))}{3 \ln (7) - \ln (5)}$

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

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

Simplify the expression.

Rewrite $- (\ln (5) - \ln (7))$ as $- 1 (\ln (5) - \ln (7))$ .

$x = \frac{- 1 (\ln (5) - \ln (7))}{3 \ln (7) - \ln (5)}$

Move the negative in front of the fraction.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.07957638 \dots$

Solve for x 7^(3x-1)=5^(x-1)

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