Simplifying Logarithms with "The Relationship"

(You might not have known that fact but, trust me, you're gonna become very familiar with it, among many others.)

How do you evaluate/simplify logs without a calculator?

To evaluate (or simplify) a logarithmic expression without a calculator (when this simplification is possible), follow these steps:

1. Set the log expression equal to a placeholder variable. For instance, when asked to simplify log₃(81), set this equal to, say, x, yielding the log equation log₃(81) = x.
2. Convert this log equation to the equivalent exponential equation. In our example, this would be the exponential equation 3^x = 81.
3. Restate the numerical value in terms of the base. Continuing our example, this means finding the power that turns 3 into 81, which is 4.
4. Equate the coefficients. In our case, this means taking 3^x = 81 = 3⁴ and equating the powers, giving us x = 4.

(Note: The above only works when the log's argument is some integer or fractional power of the base. If the argument isn't a nice power of the base [or some power of a root of the base, about which more later], then you'll have moved into calculator-based evaluation. But while you're in the section where you're doing calculator-free simplifications, it is safe to assume that every argument will indeed be come well-behaved exact power of the base.)

How can I find the power for a log expression?

If you don't know the particular power for a number that (you know from the log) has a certain base, then grab your calculator and start dividing by the base. Keep track of how many times you've divided, to find the number of factors of that base were used to get to that number.

In the case of the number 216 and a log with base 6, you'd divide 216 by 6 and get 36, which you know equals 6². This tells you that 216 = 6³.

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• Simplify log₂(8)

(Note: Setting the log expression equal to the variable y, which we found to equal the exponent on 6, illustrates how logs are exponents.

(For me, though, the "logs are powers" thing has always been more of an intellectual point than a meaningful help in understanding or working with logs. But if your instructor or textbook mentions this equivalence, the fact that y equalled the power on 2 is what they're talking about.)

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• Simplify log₅(25)

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While many of the expressions you'll be given to simplify will have whole-number values, exponents can be fractional, where fractional powers equate to roots. For instance, the power ½ on a base corresponds to the square root of that base. You will have exercises which require this knowledge.

• Simplify log₆₄(4)

This last example highlights the fact that, to be able to work intelligently with logs, you need to be pretty good with your exponents. So take the time to review them, if you're feeling a little shaky.

Sometimes, it will be difficult to figure out the exponents that you need. In many of these cases, it will be helpful to convert both the base and the argument into exponential expressions using a smaller base.

• Evaluate log₉(27)

If you're not sure of this answer, then do the evaluation of the exponential. The one-half power means the square root, so you're taking the square root of 9, which is 3, and then cubing this result, which gives you 27.

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• Simplify log₆(6)

This is always true: log_b(b) = 1 for any base b, not just for b = 6.

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• Simplify log₃(1)

This is always true: log_b(1) = 0 for any base b, not just for b = 3.

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You should expect to see "trick" questions like the folloowing on the next test:

• Simplify log₄(−16)

This is always true: log_b(a) is undefined for any negative argument a, regardless of what the base is.

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• Simplify log₂(0)

This is always true: log_b(0) is undefined for any base b, not just for b = 2.

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• Simplify log_b(b³)

This is always true: log_b(bⁿ) = n for any base b.

Some students like to think of the above simplification as meaning that the b and the log-base-b "cancel out". This is not technically correct, but it can be a useful way of thinking of things.

(Just don't say it out loud in front of your instructor.)

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• Simplify 2^log₂(9)

This last example probably looks very complicated, and can feel quite confusing. But some students view the above problem as the 2 and the log-base-2 as "cancelling out". This is not technically correct, but can be a useful way of remembering how this type of problem works.

(But, again, don't say it out loud in front of your instructor.)

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To synopsize, these are the take-aways from this lesson so far:

• The Relationship: "log_b(x) = y" means the same thing as "b^y = x".
• Logarithms are really exponents (that is, orders or powers); they're just written differently.
• log_b(b) = 1, for any base b, because b¹ = b.
• log_b(1) = 0, for any base b, because b⁰ = 1.
• log_b(a) is undefined if a is negative.
• log_b(0) is undefined for any base b.
• log_b(bⁿ) = n, for any base b.
• b^log_b(a) = a for any base b.

You might find it wise to do extra homework exercises, to make sure you have a good feel for the above facts.