Solve for x 6.1=4^(x/6-1)

$6.1 = 4^{\frac{x}{6} - 1}$

Step 1

Rewrite the equation as $4^{\frac{x}{6} - 1} = 6.1$ .

$4^{\frac{x}{6} - 1} = 6.1$

Step 2

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

$\ln (4^{\frac{x}{6} - 1}) = \ln (6.1)$

Step 3

Expand $\ln (4^{\frac{x}{6} - 1})$ by moving $\frac{x}{6} - 1$ outside the logarithm.

$(\frac{x}{6} - 1) \ln (4) = \ln (6.1)$

Step 4

Simplify the left side.

Simplify $(\frac{x}{6} - 1) \ln (4)$ .

Apply the distributive property.

$\frac{x}{6} \ln (4) - 1 \ln (4) = \ln (6.1)$

Combine $\frac{x}{6}$ and $\ln (4)$ .

$\frac{x \ln (4)}{6} - 1 \ln (4) = \ln (6.1)$

Rewrite $- 1 \ln (4)$ as $- \ln (4)$ .

$\frac{x \ln (4)}{6} - \ln (4) = \ln (6.1)$

Step 5

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

$\frac{x \ln (4)}{6} - \ln (4) - \ln (6.1) = 0$

Step 6

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

Add $\ln (4)$ to both sides of the equation.

$\frac{x \ln (4)}{6} - \ln (6.1) = \ln (4)$

Add $\ln (6.1)$ to both sides of the equation.

$\frac{x \ln (4)}{6} = \ln (4) + \ln (6.1)$

Step 7

Multiply both sides of the equation by $6$ .

$6 \frac{x \ln (4)}{6} = 6 (\ln (4) + \ln (6.1))$

Step 8

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$6 \frac{x \ln (4)}{6} = 6 (\ln (4) + \ln (6.1))$

Rewrite the expression.

$x \ln (4) = 6 (\ln (4) + \ln (6.1))$

Simplify the right side.

Apply the distributive property.

$x \ln (4) = 6 \ln (4) + 6 \ln (6.1)$

Step 9

Divide each term in $x \ln (4) = 6 \ln (4) + 6 \ln (6.1)$ by $\ln (4)$ and simplify.

Divide each term in $x \ln (4) = 6 \ln (4) + 6 \ln (6.1)$ by $\ln (4)$ .

$\frac{x \ln (4)}{\ln (4)} = \frac{6 \ln (4)}{\ln (4)} + \frac{6 \ln (6.1)}{\ln (4)}$

Simplify the left side.

Cancel the common factor of $\ln (4)$ .

Cancel the common factor.

$\frac{x \ln (4)}{\ln (4)} = \frac{6 \ln (4)}{\ln (4)} + \frac{6 \ln (6.1)}{\ln (4)}$

Divide $x$ by $1$ .

$x = \frac{6 \ln (4)}{\ln (4)} + \frac{6 \ln (6.1)}{\ln (4)}$

Simplify the right side.

Cancel the common factor of $\ln (4)$ .

Cancel the common factor.

$x = \frac{6 \ln (4)}{\ln (4)} + \frac{6 \ln (6.1)}{\ln (4)}$

Divide $6$ by $1$ .

$x = 6 + \frac{6 \ln (6.1)}{\ln (4)}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = 6 + \frac{6 \ln (6.1)}{\ln (4)}$

Decimal Form:

$x = 13.82642772 \dots$

Solve for x 6.1=4^(x/6-1)

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