Solve 10^-6(e^frac{x{26}}-1)=0.5

$$10 ^ { - 6 } ( e ^ { \frac { x } { 26 } } - 1 ) = 0.5$$

Solve for x

$$x=26\ln(500001)\approx 341.181499812$$

Steps Using Definition of Logarithm

Calculate 10 to the power of -6 and get \frac{1}{1000000}.

$$\frac{1}{1000000}\left(e^{\frac{x}{26}}-1\right)=0.5$$

Use the distributive property to multiply \frac{1}{1000000} by e^{\frac{x}{26}}-1.

$$\frac{1}{1000000}e^{\frac{x}{26}}-\frac{1}{1000000}=0.5$$

Use the rules of exponents and logarithms to solve the equation.

$$\frac{1}{1000000}e^{\frac{1}{26}x}-\frac{1}{1000000}=0.5$$

Add \frac{1}{1000000} to both sides of the equation.

$$\frac{1}{1000000}e^{\frac{1}{26}x}=\frac{500001}{1000000}$$

Multiply both sides by 1000000.

$$e^{\frac{1}{26}x}=500001$$

Take the logarithm of both sides of the equation.

$$\log(e^{\frac{1}{26}x})=\log(500001)$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$\frac{1}{26}x\log(e)=\log(500001)$$

Divide both sides by \log(e).

$$\frac{1}{26}x=\frac{\log(500001)}{\log(e)}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$\frac{1}{26}x=\log_{e}\left(500001\right)$$

Multiply both sides by 26.

$$x=\frac{\ln(500001)}{\frac{1}{26}}$$

Solve for x (complex solution)

$$x=26\ln(500001)+i\times 52\pi n_{1}$$

$$n_{1}\in \mathrm{Z}$$