Module'2'Lesson'2'
Exponential'and'Logarithmic'Functions'
Expanding'Logarithms'and'Natural'Logarithms'Video'3'Transcript'
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Hey!everyone!this!is!a!video!for!module!2!lesson!2,!exponential!and!logarithmic!
functions.!!In!this!video!we!will!expand!logarithms!and!natural!logarithms.!!!
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So!the!first!problem!says!to!expand!that!following!logarithms!or!natural!logarithms,!
that’s!your!directions.!!The!first!thing!that!I!will!expand!is!the!log!of!3!x!to!the!second!
over!7!y!to!the!third.!!So!the!first!thing!that!you!have!to!expand!here!is!going!to!be!
the!quotient.!!So!if!you!something!over!something,!it!become!the!log!of!the!
numerator!minus!the!log!of!the!denominator.!!Once!you!get!this!far!you!will!break!
up!each!piece,!so!the!log!of!3!times!x!to!the!second,!this!is!a!product,!so!that!will!
break!down!into!a!addition.!So!the!log!of!3!plus!the!log!of!x!squared.!!Now!I!am!going!
to!subtract!this!piece!of!the!problem,!so!I!am!going!to!subtract!and!this!is!a!product!
as!well,!so!its!going!to!break!down!into!a!binomial!as!well,!so!I!am!going!to!put!a!
parenthesis!here.!!So!the!log!of!7!plus!the!log!of!y!to!the!third.!!I!have!to!put!
parenthesis!there!because!there!is!a!minus!here!and!I!am!breaking!that!piece!down.!!
So!now!in!the!last!step!I!am!going!to!get!rid!of!these!parenthesis!by!distributing!a!
negative!through!as!well!as!the!exponents!at!the!same!time.!!So!if!you!have!an!
exponent,!it!comes!down!in!front!and!becomes!a!coefficient.!!So!we!have!the!log!of!3!
plus!2!times!the!log!of!x!minus!the!log!of!7,!if!I!distribute!the!negative!through,!a!
negative!through!that!would!be!a!minus,!this!exponent!of!a!3!would!come!down,!so!
it!would!e!3!times!the!log!of!y.!!So!that!would!be!that!expanded.!!!
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The!next!example!says!the!log!of!4z!to!the!fifth!over!x!minus!7.!So!the!first!thing!I!am!
going!to!do!is!make!this!a!quotient,!or!since!this!is!a!quotient!I!am!gong!to!make!this!
a!difference.!!So!it!will!be!the!log!of!the!top!minus!the!log!of!the!bottom.!!Now!there’s!
2!ways!to!break!this!piece!down,!and!I!will!do!both!ways.!!I!will!do!the!first!way!over!
here,!this!is!an!exponent!first,!I!could!bring!this!exponent!to!the!front!and!have!5!log!
of!4z!minus!and!log!of!xI7!is!not!a!product,!quotient,!and!it!doesn’t!have!and!
exponent!so!it!would!stay!the!same.!!The!last!thing,!the!next!to!last!thing!I!would!
have!to!do!is!deal!with!this!product.!!There!is!a!product!here,!so!that!would!be!5,!
since!I!am!going!to!break!this!piece!down!into!a!binomial!I!would!have!to!put!a!
parenthesis!there!(after!the!5).!!So!it!would!be!log!of!4!plus!the!log!of!z!and!then!I!
would!subtract!the!log!of!x!minus!7.!!And!that!last!thing!that!I!would!do!to!expand!
this!completely!is!to!distribute!the!5.!!So!I!would!have!5!log!of!4!plus!5!log!of!z!minus!
the!log!of!x!minus!7.!!So!all!of!these!statements!are!equivalent!statement!because!I!
used!all!the!different!rules.!
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!Another!way!to!expand!this!logarithmic!expression!is!to!use!the!quotient!rule!as!I!
did!in!the!first!way,!but!instead!of!using!the!exponent!rule!to!bring!the!5!out!in!front!
as!the!coefficient,!I!could!have!rewritten!this!first!expression!as!this,!the!log!of!4!to!
the!fifth!times!z!to!the!fifth!minus!the!log!of!x!minus!7.!!After!that,!this!logarithmic!
piece!could!have!become!the!log!of!4!to!the!fifth!plus!the!log!of!z!to!the!fifth,!which!is!
the!product!rule!minus!the!log!of!x!minus!7.!!At!that!time!you!could!make!this!
exponent!become!a!coefficient.!!So!it!would!be!5!times!the!log!of!4!plus,!bring!this!
exponent!to!the!front,!5!times!the!log!of!z!minus!the!log!of!x!minus!7.!!And!as!you!can!
see!I!expanded!that!2!different!ways,!but!I!got!the!same!thing!at!the!end.!!So!either!
way!you!do!it!is!going!to!expand!to!be!the!same!logarithmic!expression.!!
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All!right!the!third!problem!says!the!natural!log!of!m!to!the!fourth!times!n!to!the!
seventh.!!So!I!!changed!this!to!a!natural!log!in!this!example,!the!rules!are!the!same,!so!
if!this!is!a!product!it!become!the!natural!log!of!m!to!the!fourth!plus!the!natural!log!of!
n!to!the!seventh.!!And!now!to!deal!with!the!exponents,!they!come!in!front!to!become!
coefficients,!so!that!would!become!4!times!the!natural!log!of!m!plus!7!times!the!
natural!log!of!n.!!!!
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Then!the!last!problem!is!the!natural!log!of!x!squared!times!y!to!third,!all!of!that!
raised!to!the!fourth.!!So!the!first!thing!I!have!to!deal!with!is!the!exponent!because!
this!is!a!binomial!in!here!with!the!parenthesis.!!So!the!4!would!come!in!front,!so!you!
would!have!4!times!the!natural!log!of!x!squared!times!y!to!the!third.!The!only!rules!
we!have!to!work!with!are!the!product,!quotient,!and!exponent!rule.!!So!this!is!not!a!
product!or!a!quotient,!this!is!as!far!as!I!can!go!with!expanding!this.!!!
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