Solve for n 6^(3n)=43^(5n-4)

$6^{3 n} = 43^{5 n - 4}$

Step 1

Take the log of both sides of the equation.

$\ln (6^{3 n}) = \ln (43^{5 n - 4})$

Step 2

Expand $\ln (6^{3 n})$ by moving $3 n$ outside the logarithm.

$3 n \ln (6) = \ln (43^{5 n - 4})$

Step 3

Expand $\ln (43^{5 n - 4})$ by moving $5 n - 4$ outside the logarithm.

$3 n \ln (6) = (5 n - 4) \ln (43)$

Step 4

Solve the equation for $n$ .

Apply the distributive property.

$3 n \ln (6) = 5 n \ln (43) - 4 \ln (43)$

Subtract $5 n \ln (43)$ from both sides of the equation.

$3 n \ln (6) - 5 n \ln (43) = - 4 \ln (43)$

Factor $n$ out of $3 n \ln (6) - 5 n \ln (43)$ .

Factor $n$ out of $3 n \ln (6)$ .

$n (3 \ln (6)) - 5 n \ln (43) = - 4 \ln (43)$

Factor $n$ out of $- 5 n \ln (43)$ .

$n (3 \ln (6)) + n (- 5 \ln (43)) = - 4 \ln (43)$

Factor $n$ out of $n (3 \ln (6)) + n (- 5 \ln (43))$ .

$n (3 \ln (6) - 5 \ln (43)) = - 4 \ln (43)$

Divide each term in $n (3 \ln (6) - 5 \ln (43)) = - 4 \ln (43)$ by $3 \ln (6) - 5 \ln (43)$ and simplify.

Divide each term in $n (3 \ln (6) - 5 \ln (43)) = - 4 \ln (43)$ by $3 \ln (6) - 5 \ln (43)$ .

$\frac{n (3 \ln (6) - 5 \ln (43))}{3 \ln (6) - 5 \ln (43)} = \frac{- 4 \ln (43)}{3 \ln (6) - 5 \ln (43)}$

Simplify the left side.

Cancel the common factor of $3 \ln (6) - 5 \ln (43)$ .

Cancel the common factor.

$\frac{n (3 \ln (6) - 5 \ln (43))}{3 \ln (6) - 5 \ln (43)} = \frac{- 4 \ln (43)}{3 \ln (6) - 5 \ln (43)}$

Divide $n$ by $1$ .

$n = \frac{- 4 \ln (43)}{3 \ln (6) - 5 \ln (43)}$

Simplify the right side.

Move the negative in front of the fraction.

$n = - \frac{4 \ln (43)}{3 \ln (6) - 5 \ln (43)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$n = - \frac{4 \ln (43)}{3 \ln (6) - 5 \ln (43)}$

Decimal Form:

$n = 1.12017807 \dots$