Solve for k 5+6^(2k-9)=14

$5 + 6^{2 k - 9} = 14$

Step 1

Move all terms not containing $k$ to the right side of the equation.

Subtract $5$ from both sides of the equation.

$6^{2 k - 9} = 14 - 5$

Subtract $5$ from $14$ .

$6^{2 k - 9} = 9$

Step 2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (6^{2 k - 9}) = \ln (9)$

Step 3

Expand $\ln (6^{2 k - 9})$ by moving $2 k - 9$ outside the logarithm.

$(2 k - 9) \ln (6) = \ln (9)$

Step 4

Simplify the left side.

Apply the distributive property.

$2 k \ln (6) - 9 \ln (6) = \ln (9)$

Step 5

Move all the terms containing a logarithm to the left side of the equation.

$2 k \ln (6) - 9 \ln (6) - \ln (9) = 0$

Step 6

Move all terms not containing $k$ to the right side of the equation.

Add $9 \ln (6)$ to both sides of the equation.

$2 k \ln (6) - \ln (9) = 9 \ln (6)$

Add $\ln (9)$ to both sides of the equation.

$2 k \ln (6) = 9 \ln (6) + \ln (9)$

Step 7

Divide each term in $2 k \ln (6) = 9 \ln (6) + \ln (9)$ by $2 \ln (6)$ and simplify.

Divide each term in $2 k \ln (6) = 9 \ln (6) + \ln (9)$ by $2 \ln (6)$ .

$\frac{2 k \ln (6)}{2 \ln (6)} = \frac{9 \ln (6)}{2 \ln (6)} + \frac{\ln (9)}{2 \ln (6)}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 k \ln (6)}{2 \ln (6)} = \frac{9 \ln (6)}{2 \ln (6)} + \frac{\ln (9)}{2 \ln (6)}$

Rewrite the expression.

$\frac{k \ln (6)}{\ln (6)} = \frac{9 \ln (6)}{2 \ln (6)} + \frac{\ln (9)}{2 \ln (6)}$

Cancel the common factor of $\ln (6)$ .

Cancel the common factor.

$\frac{k \ln (6)}{\ln (6)} = \frac{9 \ln (6)}{2 \ln (6)} + \frac{\ln (9)}{2 \ln (6)}$

Divide $k$ by $1$ .

$k = \frac{9 \ln (6)}{2 \ln (6)} + \frac{\ln (9)}{2 \ln (6)}$

Simplify the right side.

Cancel the common factor of $\ln (6)$ .

Cancel the common factor.

$k = \frac{9 \ln (6)}{2 \ln (6)} + \frac{\ln (9)}{2 \ln (6)}$

Rewrite the expression.

$k = \frac{9}{2} + \frac{\ln (9)}{2 \ln (6)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$k = \frac{9}{2} + \frac{\ln (9)}{2 \ln (6)}$

Decimal Form:

$k = 5.11314719 \dots$

Solve for k 5+6^(2k-9)=14

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