Solve (1800)^1=2025^1frac{x{15}}

$$( 1800 ) ^ { 1 } = 2025 ^ { 1 \frac { x } { 15 } }$$

Solve for x

$$x = \frac{15 \log_{45} {(1800)}}{2} \approx 14.767940349$$

Steps Using Definition of Logarithm

Calculate 1800 to the power of 1 and get 1800.

$$1800=2025^{1\times \frac{x}{15}}$$

Express 1\times \frac{x}{15} as a single fraction.

$$1800=2025^{\frac{x}{15}}$$

Swap sides so that all variable terms are on the left hand side.

$$2025^{\frac{x}{15}}=1800$$

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

$$2025^{\frac{1}{15}x}=1800$$

Take the logarithm of both sides of the equation.

$$\log(2025^{\frac{1}{15}x})=\log(1800)$$

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

$$\frac{1}{15}x\log(2025)=\log(1800)$$

Divide both sides by \log(2025).

$$\frac{1}{15}x=\frac{\log(1800)}{\log(2025)}$$

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

$$\frac{1}{15}x=\log_{2025}\left(1800\right)$$

Multiply both sides by 15.

$$x=\frac{\log_{45}\left(1800\right)}{\frac{1}{15}\times 2}$$

Solve for x (complex solution)

$$x=\frac{15\pi n_{1}i}{\ln(45)}+\frac{15\log_{45}\left(1800\right)}{2}$$

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