Module'2'Lesson'3'
Solving'Exponential'and'Logarithmic'Equations''
Notes'Part'1'Transcript'
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Hey!everyone!this!is!a!video!for!module!2!lesson!3!solving!exponential!and!
logarithmic!equations.!
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The!examples!that!we!are!going!to!do!in!this!video,!are!solve!for!x!and!round!
answers!to!4!decimal!places.!So!if!we!look!at!number!1,!it!says!4!to!some!power!
equals!7.!!So!we!are!trying!to!get!x!all!by!itself!and!to!find!what!power!that!I!can!raise!
4!to,!to!get!7.!!So!if!you!notice!this!is!in!exponential!form!I’m!going!to!change!that!
logarithmic!form.!If!I!change!this!to!logarithmic!form,!it!becomes!the!log!base!4!of!7!
equals!x,!making!x!by!itself.!So!now!if!you!looked!at!the!notes,!you!read!about!the!
change!of!base!rule.!!If!I!have!the!log!base!4!of!7!I!can!change!that!to!be!this,!the!log!
of!the!big!number!(7)!over!the!log!of!the!base!(4).!!!So!what!that!does!it!puts!this!
logarithm!in!base!10!form,!when!I!do!that!I’m!making!it!so!I!can!plug!that!into!the!
calculator.!So!if!I!do!the!log!of!7!divided!by!the!log!of!4!I!get!that!x!is!approximately!
1.4037.!!So,!that!is!the!approximate!answer!for!x.!!To!prove!that!that!power!does!
indeed!give!me!7!when!I!raise!4!by!it,!if!you!put!in!4!raised!to!the!second!and!I!do!
answer,!its!going!to!pull!that!number!(pointing!to!the!1.403677461)!and!all!of!its!
decimal!places,!the!ones!that!are!not!even!showing,!and!raise!4!to!that!power!which!
gives!me!7.!!If!I!do!4!raised!to!1.4037,!I!get!close!to!7!but!not!exactly!7,!this!decimal!
place!is!going!to!go!on!forever.!!So!the!answer!to!number!1!is!approximately!1.4037.!!
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For!number!2,!it!gives!the!problem!in!exponential!form,!if!I!change!that!to!the!log!
base!8!of!17.3!that!would!equal!x.!!To!get!an!approximate!answer!for!x!I!could!do!the!
log!of!17.3!over!the!log!of!8.!!!Put!that!in!my!calculator,!remember!to!close!the!
parenthesis!around!each!log!there,!so!the!log!of!17.3!divided!by!the!log!of!8!and!the!
approximate!answer!for!number!8!would!be!1.3709.!
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For!the!next!one,!“e”!raised!to!some!power!equals!13.!So!read!in!your!notes,!if!you!
use!“e”!you!know!the!inverse!of!“e”!is!going!to!be!the!natural!logarithm.!!So!if!I!do!the!
natural!logarithm!of!13!that!would!equal!x.!!Note,!I!don’t!have!to!put!“e”!as!the!small!
base!here!because!if!I!don’t!put!it!and!it’s!a!natural!logarithm!its!understood!to!be!
“e.”!!so!now!x!is!all!by!itself,!so!now!this!is!the!exact!answer!for!x,!to!get!he!
approximate!answer!use!your!calculator!and!do!the!natural!log!of!13!and!hit!enter.!!
So!x!is!going!to!be!approximately!2.5649.!!Now!what!would!have!happened!if!I!
would!have!put!a!base!“e”!there?!!Let’s!do!that!in!the!calculator,!so!if!I!do!the!natural!
log!of!13!divided!by,!because!this!would!have!become!the!natural!of!of!13!over!the!
natural!log!of!e,!do!I!can!do!divided!by!the!natural!log!of!“e.”!!If!I!do!second!“e”!that!
would!give!me!the!same!thing,!because!once!again!if!I!don’t!put!that,!its!understood!
so!the!calculator!knows!to!do!that.!!So!that!is!equivalent!to!the!natural!log!of!13.!
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All!right,!solve!number!4!this!gives!a!logarithmic!form!and!I!am!trying!to!get!this!x!
by!itself.!!!So!let’s!change!that!form!to!exponential!form,!so!this!would!be!equivalent!
to!3!to!the!fifth!power!would!equal!x.!!So!now!all!you!have!to!do!to!solve!for!x!is!to!
put!in!3!raised!to!the!fifth!power.!!So!x!of!course!is!going!to!be!exactly!243.!
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Moving!on!to!number!5,!we!have!log!base!x!of!7!equals!4.!!So!our!goal!is!to!get!x!by!
itself.!!So!if!you!notice,!I!can!change!this!to!exponential!form,!x!to!the!fourth!power!
has!to!equal!7.!!So!how!do!I!get!x!by!itself?!!I!can!either!take!the!4th!root!of!that!or!an!
equivalent!would!be!raise!it!to!the!one!fourth!power.!!If!I!raise!the!left!side!to!the!
one!fourth!power,!I!raise!the!right!side!to!the!one!fourth!power.!So!this!makes!this!x!
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on!that!side!and!then!7!raised!to!the!one!fourth!would!be!approximately,!1.6266.!!Oh!
that!got!messy,!let!me!do!that!again.!!1.6266.!!And!that!would!be!the!approximate!
answer.!!!
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Number!6!says!to!solve!for!x!and!notice!that!x!is!already!by!itself,!to!get!the!
approximate!answer!for!x,!you!would!just!need!to!know!the!change!of!base!formula,!!
and!do!the!log!o!17.5!over!the!log!of!5,!and!that!equals!x.!!So!if!I!put!that!in!the!
calculator,!log!of!17.5!divided!by!the!log!of!5,!I!would!get!he!answer!x!is!
approximately!1.7784.!
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Number!7,!says!the!natural!log!of!x!equals!5,!so!now!I!have!to!get!this!x!by!itself.!So!
remember!if!you!have!the!natural!log!and!no!base!that!means!that!“e”!is!understood!
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to!be!the!base.!!So!I!could!change!this!natural!log!form!to!exponential!form!by!doing!
“e”!to!the!fifth!power!would!equal!my!x!value.!!So!to!do!that!I!would!do!second!“e,”!
you!could!also!use!the!one!on!the!natural!log.!!If!you!do!the!second!“e”!on!the!
division(meaning!the!division!button!on!the!calculator),!you!actually!have!to!use!the!
carat!button!and!raise!it!to!the!fifth!power.!!If!you!use!the!one!on!the!natural!log!it!
automatically!raises!it!to!a!power!and!puts!a!parenthesis!and!you!could!just!close!
the!parenthesis.!!It’s!going!to!give!you!the!same!number!either!way.!!!So!the!
approximate!answer!to!x!for!number!7!would!be,!x!is!approximately!148.4132.!
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Just!as!with!natural!log,!if!I!just!have!a!log!my!understood!base!is!10.!!So!for!number!
8!to!get!that!x!by!itself!I!could!rewrite!that!to!be!10!raised!to!the!89!power!is!going!
to!be!x.!!And!that’s!going!to!be!a!really!big!number.!!So!if!I!do!10!raised!to!the!89th!
power,!I!get!1!E!89.!!So!how!do!I!write!that?!!So!if!you!remember!this!is!scientific!
notation!that!I!can!use!here.!!So!x!would!be!1.00!times!10!to!the!89th!power.!!So!I!
would!move!that!decimal!place!89!times!to!the!right.!!
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So!to!solve!these!types!of!equations,!all!I!had!to!do!was!to!change!from!one!form!to!
another!and!use!the!change!of!base.!!
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