Properties of Logarithms --Part II

Mastering algebraic properties of logarithms requires much practice. To help you along, here is a list of basic properties.

Properties of Logarithms

1) log[a](a^x) = x , a^log[a](x) = x

2) log[a](AB) = log[a](A)+log[a](B)

3) log[a](A/B) = log[a](A)-log[a](B)

4) log[a](A^p) = p*log[a](A)

5) log[a](x) = ln(x)/ln(a)

Problem 1. Express in terms of sums and differences of logarithms.

log[a](x^2*y^5/(z^4+1)) .

(A) 2*log[a](x)+5*log[a](y)-4*log[a](z)+log[a](1)

(B) 2*log[a](x)+5*log[a](y)-4*log[a](z)-log[a](1)

(C) 2*log[a](x)+5*log[a](y)-log[a](z^4+1)

(D) 2*log[a](x)+5*log[a](y)-4*log[a](z)

CLICK HERE FOR SOLUTION OF PROBLEM 1

Problem 2. Simplify

exp(2*ln(sqrt(x))+4*ln(x)-3*ln(x^2)) .

(A) 2*ln(sqrt(x))+4*ln(x)-3*ln(x^2)

(B) 2*sqrt(x)+4*x-3*x^2

(C) 8*sqrt(x)/(3*x)

(D) 1/x

CLICK HERE FOR SOLUTION OF PROBLEM 2

Problem 3. Express in terms of sums and differences of logarithms.

log[2](sqrt(x^5*y/(z^7*w))) .

(A) 5*log[2](x)+log[2](y)-7*log[2](z)+log[2](w)

(B) 5*log[2](x)/2+log[2](y)/2-7*log[2](z)/2+log[2](w)/2...

(C) 5*log[2](x)/2+log[2](y)/2-7*log[2](z)/2-log[2](w)/2...

(D) 3*log[2](x)+3*log[2](y)-4*log[2](z)-4*log[2](w)

Problem 4. Simplify

ln(exp(x)/(x^3)) .

(A) x-3*ln(x) (B) e-3*ln(x) (C) e-2*ln(x) (D) x+ln(x^3)

Problem 5. Simplify

exp(3*ln(x+y)) .

(A) 3*(x+y) (B) Can't be simplified (C) x^3+y^3 (D) (x+y)^3

Problem 6. Simplify

ln(exp(x)*exp(2*y)) .

(A) x+2*y (B) 2*(x+y) (C) x+y^2 (D) exp(x)*exp(2*y)

Precalculus Tutorials, B. Kaskosz and L. Pakula, 2002.