Solve 10.5^x=1/3

$$10.5 ^ { x } = \frac { 1 } { 3 }$$

Solve for x

$$x=-\log_{10.5}\left(3\right)\approx -0.467221166$$

Steps Using Definition of Logarithm

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

$$10.5^{x}=\frac{1}{3}$$

Take the logarithm of both sides of the equation.

$$\log(10.5^{x})=\log(\frac{1}{3})$$

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

$$x\log(10.5)=\log(\frac{1}{3})$$

Divide both sides by \log(10.5).

$$x=\frac{\log(\frac{1}{3})}{\log(10.5)}$$

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

$$x=\log_{10.5}\left(\frac{1}{3}\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(10.5)}-\log_{10.5}\left(3\right)$$

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