8-4 Properties of Logarithms

Use the change of base formula to rewrite and evaluate logs

Use properties of logs to evaluate or rewrite log expressionsUse properties of logarithms to expand or condense

logarithmic expressionsUse logarithmic functions to model and solve real-life

problems.

Properties of Logarithms

• Product Property: loga (uv) = loga u + loga v

• Quotient Property: loga (u/v) = loga u - loga v

• Power Property: loga un = n loga u

Using Properties of Logs to find the exact value of the expression

Examplelog5 35

ln e6 – ln e2

Rewrite--log5 (5)1/3

Bring exponent out front.1/3 log5 (5)

= 1/3

Bring exponents out front.

6ln e – 2ln e

So--

6 – 2 = 4

OR we could have rewritten this as division—

Ln e6 = lne4 = 4lne = 4 e2

Using Properties of Logarithms to expand the expression as a sum,

difference and/or constantln 2/27 = ln 2 - ln 27

log310z = log310 + log3z

ln 6

x2 + 1

log 4x2y = log 4 + log x2 + log y

= log 4 + 2log x + log y

= ln 6 – ln (x2 + 1)1/2

= ln 6 – 1/2ln (x2 + 1)

Write the expression as a single logarithm (Go Backwards)

ln y + ln t

= ln yt log 8 – log t = log 8/t

-4ln 2xt = ln (2xt)-4

2 ln 8 + 5 ln (x – 4) = ln 82 + ln (x – 4)5

= ln 82(x – 4)5

1/3[log x + log (x + 1)] =[log x(x + 1)]1/3

2[3ln x – ln (x + 1) – ln(x – 1)] =[3ln x – ln (x + 1) – ln(x – 1)]2

= ln x3 2

(x + 1)(x – 1) Foil this

• Good problems to assign—

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