Properties of Logarithms and Exponents

Laws of Logarithms

1. $\log_b(xy)=\log_b(x)+\log_b(y)$

2. $\log_b(\frac{x}{y})=\log_b(x)-\log_b(y)$

3. $\log_b(x^r)=r(\log_b x)$

Laws of Exponents

1. $(b^t)(b^u)=b^{t+u}$

2. $\frac{b^t}{b^u}=b^{t-u}$

3. $(b^t)^u=b^{tu}$

Ex 1 Simplify the expression $\frac{(x^2)^4 x^5}{x^4 (x^3)^2}$.

Sol $\frac{(x^2)^4 x^5}{x^4 (x^3)^2}=\frac{x^8 x^5}{x^4 x^6}$ $=\frac{x^{13}}{x^{10}}=x^3$.

Ex 2 Simplify the expression $\ln((x+5)^3 \sqrt{2x-7})$.

Sol $\ln((x+5)^3 \sqrt{2x-7})=\ln((x+5)^3)+\ln(\sqrt{2x-7})$ $=\ln((x+5)^3)+\ln((2x-7)^\frac{1}{2})=3\ln(x+5)+\frac{1}{2}\ln(2x-7)$.

Pr A Find all solutions of $x^{2/3}=4$.

Pr B Simplify the expression

$$\ln\left(\frac{x^4}{(2x-1)^3(7x-5)^8}\right)$$

.

Pr C Use properties of logarithms to rewrite $2\ln(3x-4)-5\ln(2x-7)$ as the logarithm of a single expression.

Pr 1 Simplify the expression

$$\frac{x^3 (x^4)^5}{x^7 (x^2)^4}$$

.

Pr 2 Simplify the expression

$$\ln\left(\frac{(x^2+4)^5 \sqrt[3]{4x-3}}{\sqrt{3x-5} (7x+2)^9}\right)$$

.

Pr 3 If $\log_b x=2.3$ and $\log_b y=3.1$, find $\log_b (\frac {b^5 x^2}{y^3})$.

Pr 4 Use properties of logarithms to rewrite

$$1/2\log_5(4x-3)-1/2\log_5(3x+1)-3/2\log_5(x^2+6)$$

as the logarithm of a single expression.

Pr 5 Simplify the expression $(\log_9 3)(\log_5 1/25)$.

Pr 6 Simplify the expression $\ln(\ln e)+\log_2 8$.

Pr 7 Simplify the expression $\log_4 \frac{36}{5}+\log_4 \frac{10}{9}$.

Pr 8 If $\ln x =r$ and $\ln b=s$, find $\log_b x$.

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