The logs rules work "backwards", so you can condense ("compress"?) strings of log expressions into one log with a complicated argument. When they tell you to "simplify" a log expression, this usually means they will have given you lots of log terms, each containing a simple argument, and they want you to combine everything into one log with a complicated argument. "Simplifying" in this context usually means the opposite of "expanding".
There is no standard definition, in this context, for "simplifying". You have to use your own good sense. If they give you a big complicated thing and ask you to "simplify", then they almost certainly mean "expand". If they give you a string of log terms and ask you to "simplify", then they almost certainly mean "condense".
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Let's see how condensing log expressions works.
Since these logs have the same base, the addition outside can be turned into multiplication inside:
log_{2}(x) + log_{2}(y) = log_{2}(xy)
Then the answer is:
Since these logs have the same base, the subtraction outside can be turned into division inside:
log_{3}(4) – log_{3}(5) = log_{3}(^{4}/_{5})
Then my answer is:
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The multiplier out front can be taken inside as an exponent:
2 · log_{3}(x) = log_{3}(x^{2})
Then my final answer is:
I will get rid of the multipliers by moving them inside as powers:
3log_{2}(x) – 4log_{2}(x + 3) + log_{2}(y)
= log_{2}(x^{3}) – log_{2}((x + 3)^{4}) + log_{2}(y)
Then I'll put the added terms together, moving the one "minus" term to the end of the string:
log_{2}(x^{3}) – log_{2}((x + 3)^{4}) + log_{2}(y)
= log_{2}(x^{3}) + log_{2}(y) – log_{2}((x + 3)^{4})
...and convert the addition outside to multiplication inside:
log_{2}(x^{3}) + log_{2}(y) – log_{2}((x + 3)^{4})
= log_{2}(x^{3}y) – log_{2}((x + 3)^{4})
Then I'll account for the subtracted term by combining it inside with division:
Then my final answer is:
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Pay particular attention to how I grouped the log terms according to sign. This can be very important, and is where many students get lost and then lose points. Don't try to convert addition outside to multiplication inside, or subtraction outside to division inside, until you've made sure that all the "plus" terms are together up front, followed by all the "minus" terms. Then you can combine by multiplication within each set, and then finish by converting the big "minus" that's subtracted from the big "plus" into one big division inside one log.
You can use the Mathway widget below to practice simplifying a logarithmic expression. Try the entered exercise, or type in your own exercise. Then click the button to compare your answer to Mathway's. (Or skip the widget and continue with the lesson.)
(Clicking on "Tap to view steps" on the widget's answer screen will take you to the Mathway site for a paid upgrade.)
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