Solve for x 6.1=1.085^(0.01x)+1

$6.1 = {1.085}^{0.01 x} + 1$

Rewrite the equation as ${1.085}^{0.01 x} + 1 = 6.1$ .

${1.085}^{0.01 x} + 1 = 6.1$

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

Subtract $1$ from both sides of the equation.

${1.085}^{0.01 x} = 6.1 - 1$

Subtract $1$ from $6.1$ .

${1.085}^{0.01 x} = 5.1$

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

$\ln ({1.085}^{0.01 x}) = \ln (5.1)$

Expand the left side.

Expand $\ln ({1.085}^{0.01 x})$ by moving $0.01 x$ outside the logarithm.

$0.01 x \ln (1.085) = \ln (5.1)$

Multiply $\ln (1.085)$ by $0.01$ .

$0.00081579 x = \ln (5.1)$

Divide each term in $0.00081579 x = \ln (5.1)$ by $0.00081579$ and simplify.

Divide each term in $0.00081579 x = \ln (5.1)$ by $0.00081579$ .

$\frac{0.00081579 x}{0.00081579} = \frac{\ln (5.1)}{0.00081579}$

Simplify the left side.

Cancel the common factor of $0.00081579$ .

Cancel the common factor.

$\frac{0.00081579 x}{0.00081579} = \frac{\ln (5.1)}{0.00081579}$

Divide $x$ by $1$ .

$x = \frac{\ln (5.1)}{0.00081579}$

Simplify the right side.

Replace $e$ with an approximation.

$x = \frac{\log_{2.71828182} (5.1)}{0.00081579}$

Log base $2.71828182$ of $5.1$ is approximately $1.62924053$ .

$x = \frac{1.62924053}{0.00081579}$

Divide $1.62924053$ by $0.00081579$ .

$x = 1997.10811412$