Solve for x 3.8=1.1^(x-5)

$3.8 = {1.1}^{x - 5}$

Step 1

Rewrite the equation as ${1.1}^{x - 5} = 3.8$ .

${1.1}^{x - 5} = 3.8$

Step 2

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

$\ln ({1.1}^{x - 5}) = \ln (3.8)$

Step 3

Expand $\ln ({1.1}^{x - 5})$ by moving $x - 5$ outside the logarithm.

$(x - 5) \ln (1.1) = \ln (3.8)$

Step 4

Simplify the left side.

Apply the distributive property.

$x \ln (1.1) - 5 \ln (1.1) = \ln (3.8)$

Step 5

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

$x \ln (1.1) - 5 \ln (1.1) - \ln (3.8) = 0$

Step 6

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

Add $5 \ln (1.1)$ to both sides of the equation.

$x \ln (1.1) - \ln (3.8) = 5 \ln (1.1)$

Add $\ln (3.8)$ to both sides of the equation.

$x \ln (1.1) = 5 \ln (1.1) + \ln (3.8)$

Step 7

Divide each term in $x \ln (1.1) = 5 \ln (1.1) + \ln (3.8)$ by $\ln (1.1)$ and simplify.

Divide each term in $x \ln (1.1) = 5 \ln (1.1) + \ln (3.8)$ by $\ln (1.1)$ .

$\frac{x \ln (1.1)}{\ln (1.1)} = \frac{5 \ln (1.1)}{\ln (1.1)} + \frac{\ln (3.8)}{\ln (1.1)}$

Simplify the left side.

Cancel the common factor of $\ln (1.1)$ .

Cancel the common factor.

$\frac{x \ln (1.1)}{\ln (1.1)} = \frac{5 \ln (1.1)}{\ln (1.1)} + \frac{\ln (3.8)}{\ln (1.1)}$

Divide $x$ by $1$ .

$x = \frac{5 \ln (1.1)}{\ln (1.1)} + \frac{\ln (3.8)}{\ln (1.1)}$

Simplify the right side.

Cancel the common factor of $\ln (1.1)$ .

Cancel the common factor.

$x = \frac{5 \ln (1.1)}{\ln (1.1)} + \frac{\ln (3.8)}{\ln (1.1)}$

Divide $5$ by $1$ .

$x = 5 + \frac{\ln (3.8)}{\ln (1.1)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = 5 + \frac{\ln (3.8)}{\ln (1.1)}$

Decimal Form:

$x = 19.00690953 \dots$