# Sec. 5.3 Properties of Logarithms

Sec. 5.3 Properties of Logarithms Solve for x log230 = x 2x = 30 30 cant be broken down to a power of 2.

However, since we know 24 = 16 and 25 = 32 and 30 is between 16 and 32 x is between 4 & 5 This is estimated! To find exact answers use change of base formula

log b x log a x log b a or ln x log a x ln a

Ex. 1 Use change of base formula a) log430 b) log214 You can change to any base but to use the calculator you must change the base to 10 or the natural base e

( log key or ln key) Ex. 2 evaluate using ln a) log430 b) log214 Evaluate 1) log74

2) log40.55 3)log0.015 Properties of Logarithms 1) loga(uv) = logau + logav (like the property of exponents where you add the exp. if they have the same

base.) 2) loga(u/v) = logau logav (Comes from the division of exponents) 3) logaun = n logau These properties are also true for natural logs. Applying these properties is also called expanding the logarithmic

expression xy ln 2 4 Ex. 3 Write in terms of ln 2 and ln 3

a) ln 6 b) ln 2/27 Ex. a) log3(92 43) b) log2 (8/11) c) ln(e3/5)

Condensing Logarithmic Expressions Ex.7 log10x +3 log10 (x + 1)

Ex.8 2 ln (x + 2) ln x Ex. a) ln (x + 2) ln 8 b) log45 + log4x c) 8 log3 (x 1) d) 3 log45 + log48 e) ln x + ln 3 ln (x + 2)

f) 5 log6 (x + 2) 3 log6 x

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