Solve for x 5^(2x+3)=3^(x-1)

$5^{2 x + 3} = 3^{x - 1}$

Step 1

Take the log of both sides of the equation.

$\ln (5^{2 x + 3}) = \ln (3^{x - 1})$

Step 2

Expand $\ln (5^{2 x + 3})$ by moving $2 x + 3$ outside the logarithm.

$(2 x + 3) \ln (5) = \ln (3^{x - 1})$

Step 3

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

$(2 x + 3) \ln (5) = (x - 1) \ln (3)$

Step 4

Solve the equation for $x$ .

Simplify $(2 x + 3) \ln (5)$ .

Rewrite.

$0 + 0 + (2 x + 3) \ln (5) = (x - 1) \ln (3)$

Simplify by adding zeros.

$(2 x + 3) \ln (5) = (x - 1) \ln (3)$

Apply the distributive property.

$2 x \ln (5) + 3 \ln (5) = (x - 1) \ln (3)$

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

Apply the distributive property.

$2 x \ln (5) + 3 \ln (5) = x \ln (3) - 1 \ln (3)$

Rewrite $- 1 \ln (3)$ as $- \ln (3)$ .

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

Subtract $x \ln (3)$ from both sides of the equation.

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

Subtract $3 \ln (5)$ from both sides of the equation.

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

Factor $x$ out of $2 x \ln (5) - x \ln (3)$ .

Factor $x$ out of $2 x \ln (5)$ .

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

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

$x (2 \ln (5)) + x (- 1 \ln (3)) = - \ln (3) - 3 \ln (5)$

Factor $x$ out of $x (2 \ln (5)) + x (- 1 \ln (3))$ .

$x (2 \ln (5) - 1 \ln (3)) = - \ln (3) - 3 \ln (5)$

Rewrite $- 1 \ln (3)$ as $- \ln (3)$ .

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

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

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

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

Simplify the left side.

Cancel the common factor of $2 \ln (5) - \ln (3)$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Combine the numerators over the common denominator.

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

Factor $- 1$ out of $- \ln (3)$ .

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

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

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

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

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

Simplify the expression.

Rewrite $- (\ln (3) + 3 \ln (5))$ as $- 1 (\ln (3) + 3 \ln (5))$ .

$x = \frac{- 1 (\ln (3) + 3 \ln (5))}{2 \ln (5) - \ln (3)}$

Move the negative in front of the fraction.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 2.79537233 \dots$

Solve for x 5^(2x+3)=3^(x-1)

Related Questions

Support LCMGCF.com by disabling your adblocker.