Solve 5469.78=4500*1.05^n

$$5469.78 = 4500 \times 1.05 ^ { n }$$

Solve for n

$$n=\log_{1.05}\left(\frac{91163}{75000}\right)\approx 4.000007026$$

Steps Using Definition of Logarithm

Divide both sides by 4500.

$$\frac{5469.78}{4500}=1.05^{n}$$

Expand \frac{5469.78}{4500} by multiplying both numerator and the denominator by 100.

$$\frac{546978}{450000}=1.05^{n}$$

Reduce the fraction \frac{546978}{450000} to lowest terms by extracting and canceling out 6.

$$\frac{91163}{75000}=1.05^{n}$$

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

$$1.05^{n}=\frac{91163}{75000}$$

Take the logarithm of both sides of the equation.

$$\log(1.05^{n})=\log(\frac{91163}{75000})$$

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

$$n\log(1.05)=\log(\frac{91163}{75000})$$

Divide both sides by \log(1.05).

$$n=\frac{\log(\frac{91163}{75000})}{\log(1.05)}$$

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

$$n=\log_{1.05}\left(\frac{91163}{75000}\right)$$