Properties of Logarithms
1. Properties of Logarithms
2. Applying Properties of Logarithms
3. Condensing Logarithmic Expressions to Single Logarithms
4. Solving Logarithms With a Base that is not 10 or e

Objective 1: Work with Properties of Logarithms

Properties of Logarithms

*loga is assumed to be base 10



The Log of a Product Equals the Sum of the Logs:


The Log of a Quotient Equals the Difference of the Logs


The Log of a Power Equals the Product of the Power and the Log


logaak=x; k=x

Natural Logarithms: ln

  • Natural logarithm is always base “e”
  • lne=1
  • logea=lna

Objective 2: Applying Properties of Logarithms

Use properties of logarithms to write each expression as a sum, difference, and/or constant multiple of logarithms

1. Using “log(ab)=loga+logb” Property

a. log7t → log7 + logt
b. ln(xyz) → ln(x)+ln(y)+ln(z)
c. log2(xy) → log2(x)+log2(y)

2. Using “logab=bloga” Property

a. log6t6 → 6log6(x)
b. ln(√x) → ln(x)½ → ½ln(x)
c. ln(3√x) → ln(x) → ⅓ln(x)
d. log2t-5 → -5log2(t)

3. Using “logaa=1” Property

a. log1010x→ log10(x)+1

4. Using “log(a/b)=loga-logb” Property

a. log10(8/n) → log108-log10n
b. ln(y/x) → ln(y)-ln(x)
c. log(n/t) → log(n)-log(t)

Using Multiple Properties
a. ln(x2-1)/(x2) → ln(x2-1)-2ln(x)

b. ln(x4[√y])/(z4) → (½ln(y)+4ln(x))-4ln(z)

c. ln((z2)/(y3)) → 2ln(z)-3ln(y)

d. ln[z(z-1)2] → ln(z)+ln(z-1)2 → ln(z)+2ln(z-1)

e. logb((x2)/(y3z2)) → 2logb(x)-(3logb(y)+2logb(z))

Objective 3: Condensing Logarithmic Expressions to Single Logarithms

Logarithmic expressions can be condensed into single logarithms using properties of logs.

Write each expression as a single logarithm whose coefficient is one
1. ln(x)-ln(y) → ln(x/y)

2. ln(x)+½ln(x2+1) → ln(x)+ln(x2+1)½ → ln(x√(1+x2))

3. log2a-2log2b → log2a-log2b2→ log2(a/b2)

4. logu2-logv→ log(u2/v)

5. 2logx+⅓logy-3logz → log((x2*3√(y))/z3)

Objective 4: Solving Logarithms with a Base That is not 10 or e
Approximate y=log312

- 3y=12 *change to exponential form
- ln3y=ln12 *take ln of both sides
- yln3=ln12 *use properties of logs to bring down the exponent
- y(ln3)/(ln3)=(ln12)/(ln12) *isolated the variable
- y=(ln12)/(ln3)
- y≈2.261859 *find the decimal equivalent using a calculator

Practice Problems
Using properties of logarithms, expand the following:
a. log3(x/9)
b. log2(z4/(t5m3))
c. log4z7

Write the following expressions as a single logarithm:
a. 3log5u+4log5v
b. ln(e)-ln(x)
c. ½log5(x)-⅓log5(y)

Solve the following:
a. y=log321
b. y=log518
c. y=log422