Solve 10^2*6^x+y=3(4*30)^2

$$10 ^ { 2 } \cdot 6 ^ { x + y } = 3 ( 4 \cdot 30 ) ^ { 2 }$$

Solve for x

$$x=-\frac{\ln(2)y+\ln(3)y+\ln(\frac{1}{432})}{\ln(6)}$$

Steps Using Definition of Logarithm

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

$$100\times 6^{x+y}=43200$$

Divide both sides by 100.

$$6^{x+y}=432$$

Take the logarithm of both sides of the equation.

$$\log(6^{x+y})=\log(432)$$

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

$$\left(x+y\right)\log(6)=\log(432)$$

Divide both sides by \log(6).

$$x+y=\frac{\log(432)}{\log(6)}$$

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

$$x+y=\log_{6}\left(432\right)$$

Subtract y from both sides of the equation.

$$x=\log_{6}\left(432\right)-y$$

Solve for y

$$y=-\frac{\ln(2)x+\ln(3)x+\ln(\frac{1}{432})}{\ln(6)}$$

Steps Using Definition of Logarithm

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

$$100\times 6^{y+x}=43200$$

Divide both sides by 100.

$$6^{y+x}=432$$

Take the logarithm of both sides of the equation.

$$\log(6^{y+x})=\log(432)$$

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

$$\left(y+x\right)\log(6)=\log(432)$$

Divide both sides by \log(6).

$$y+x=\frac{\log(432)}{\log(6)}$$

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

$$y+x=\log_{6}\left(432\right)$$

Subtract x from both sides of the equation.

$$y=\log_{6}\left(432\right)-x$$