Solve 5000(1.3^0.25x)=7500

$$5000( { 1.3 }^{ 0.25x } )=7500$$

Solve for x

$$x=-\frac{4\ln(\frac{2}{3})}{\ln(13)-\ln(10)}\approx 6.181712421$$

Steps Using Definition of Logarithm

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

$$5000\times 1.3^{0.25x}=7500$$

Divide both sides by 5000.

$$1.3^{0.25x}=\frac{3}{2}$$

Take the logarithm of both sides of the equation.

$$\log(1.3^{0.25x})=\log(\frac{3}{2})$$

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

$$0.25x\log(1.3)=\log(\frac{3}{2})$$

Divide both sides by \log(1.3).

$$0.25x=\frac{\log(\frac{3}{2})}{\log(1.3)}$$

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

$$0.25x=\log_{1.3}\left(\frac{3}{2}\right)$$

Multiply both sides by 4.

$$x=\frac{\ln(\frac{3}{2})}{0.25\ln(\frac{13}{10})}$$