{"id":6534,"date":"2016-04-12T10:33:57","date_gmt":"2016-04-12T17:33:57","guid":{"rendered":"https:\/\/magoosh.com\/act\/?p=6534"},"modified":"2022-08-08T10:24:22","modified_gmt":"2022-08-08T17:24:22","slug":"logarithms-tips-for-act-math","status":"publish","type":"post","link":"https:\/\/magoosh.com\/act\/logarithms-tips-for-act-math\/","title":{"rendered":"ACT Math Logarithms: What You Need to Know"},"content":{"rendered":"

Logarithms are not incredibly common on the ACT, but you are likely to see one, or maybe two, amongst the harder problems on the ACT Math test<\/a>, so if you are shooting for a top score on ACT Math<\/a>, logarithms are worth knowing. <\/p>\n

First of all\u2026<\/p>\n

What is a Logarithm?<\/h2>\n

A logarithm is the power to which a number must be raised in order to produce some other number. <\/p>\n

If you went, \u201cHuh?\u201d I don\u2019t blame you. <\/p>\n

But you probably know about exponents:<\/strong><\/p>\n

If we see 32<\/sup>, we know this is 3 x 3, which equals 9.<\/p>\n

If we see 45<\/sup>, we know this is 4 x 4 x 4 x 4 x 4, which equals 1024.<\/p>\n

Logarithms are all about thinking about exponents in a different way.<\/em> <\/strong><\/p>\n

Here\u2019s the question logarithms answer<\/strong>: What is the power something is raised to in order to get a number that we know?<\/p>\n

So let\u2019s say we are trying to figure out what power 3 must be raised to to get 9. Well, we just figured that out above: it\u2019s 2. And what if we are trying to figure out what power 4 must be raised to in order to get 1024? Again, we saw this above, it\u2019s 5. <\/p>\n

This brings us to the mathematical definition of a logarithm:
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Definition of a Logarithm<\/h2>\n

If loga<\/sub>b = c, then ac<\/sup> = b<\/p>\n

So you need to remember, \u201cWhat power do I need to raise a to to get b<\/strong><\/em>? <\/p>\n

Using our example above, what does log4<\/sub>1024 = ______? <\/p>\n

Hopefully, you said 5. <\/p>\n

Now try this one:<\/strong><\/p>\n

What is log2<\/sub>64? <\/p>\n

(think to yourself what power do I need to raise 2 to to get 64?)<\/p>\n

The answer is 6. 26<\/sup> = 64. <\/p>\n

A Quick Recap<\/h2>\n

If you find logarithms super-strange looking, you\u2019re definitely not alone! As we’ve seen, we\u2019re used to working with exponents in a format like y = xa<\/sup>. In \u201clogs\u201d that equation is equal to logx<\/sub>(y) = a. Or, looking at an example with actual numbers: 32<\/sup> = 9 is the equivalent of log3<\/sub>(9) = 2<\/p>\n

We would read the logarithm out loud as \u201clog-base 3 of <\/i>9 equals 2.\u201d A helpful way to remember this is to notice that whatever is on the other side of the equals sign is the exponent, and that the tiny number is the exponent base.<\/p>\n

Change of Base Rule<\/h2>\n

If you have a scientific or graphing calculator, your calculator has a log button. But this log button only calculates bases of ten (log10<\/sub>). So one more important trick to remember about logs that will help you quickly convert logs of any base to ones you can plug in your calculator is the change of base rule. Here is is:<\/p>\n

Logb<\/sub>a = log a \/ log b <\/strong> (a base of 10 is implied when it is not written). <\/p>\n

So if you see log4<\/sub>8, you can convert this to log 8 \/ log 4 and plug this in your calculator to get the answer 3\/2. Which if you check it back in the problem, it makes sense: 43\/2<\/sup> = 8. <\/p>\n

For more info on mastering logarithms, check out the video below!
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