Logarithms

1st- If M and N are positive real numbers and b is a positive number other than 1,log b MN= log b M+ log b N.

2nd- is if log b M/N=log b M-log b N.

3rd- is log b M= log b N if and only if M=N.

4th- log b M^k=k log b M for any real number k. Use this rule when your trying to solve an exponent.

Example 1:

Expand:

ln x - (1/2) ln (x² + 1)

Example 2:

Condense: 3 log x + 2 log y - (1/2) log z

Example 3:

Expand:

Log₅ xy² =

Log₅ x + Log₅ y² = Log₅ x + 2 Log₅ y

Example 4:

Write as single:

Log₂ (x + 1) + Log₂ 3 = 4

Log₂ 3(x + 1) = 4

3(x + 1) = 2⁴

3x + 3 = 16
3x = 13
x = 13/3

Example 5:

Expand:

Log₇ (xy/z²) =
Log₇ x + Log₇ y - 2 Log₇ z

Example 6:

Write as single:

1) Log x + Log y - Log z =

Log (xy)/z

2) 2 Ln x + 3 Ln y =

Ln x²y³

3) (1/2) Ln x - (1/3) Ln y =