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For the equation $2^x = 7$

The textbook says to use log base ten to solve it like this $\log 2^x = \log 7$.

I then re-arrange it so that it reads $x \log 2 = \log 7$ then divide the RHS by $\log 2$ to isolate the $x$. I understand this part.

I can alternatively solve it in an easier way by simply using $\log_2 7$ on my calculator.

Using both methods the answer comes to the same which is $2.807$

My question is twofold:

  1. Why would the textbook suggest to use log base ten rather than simply using log base two?

  2. I can see how using log base ten and the suggested method in the textbook makes me arrive at the same answer but I don't understand WHY this is so. How does base ten play a factor in the whole scope of things.

Thank you

Henry
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Y.M.80
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    Logarithm base 10 is a natural choice for computations simply because it's easy to find on most calculators. But otherwise, I don't see any particular reason. – Matti P. Feb 22 '18 at 07:48
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    In other words, I think most calculators don't have a direct way of computing $\log_a(x)$ for a given base $a$. It has to be calculated $\log_a(x) = \frac{\log_{10}(x)}{\log_{10}(a)}$. You must have a fancy calculator. – Matti P. Feb 22 '18 at 07:50
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    It's actually a smartphone app that serves as a virtual scientific calculator :) – Y.M.80 Feb 22 '18 at 08:52
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    The textbook method has the additional advantage of working in any base. For example, many calculators and software packages call the natural logarithm $\log$ rather than $\ln$ - don't assume that $\log$ without a base means $\log_{10}$! – Sanchises Feb 22 '18 at 14:34
  • Hint: $\log_2 7$ is equal to $\log_2 7 / \log_2 2$ -- does that give you any insight? – Eric Lippert Feb 22 '18 at 16:20
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    My take on this fact is that you only need one logarithm; the rest (that is, the ones with other bases) are just multiples of that one. Which one your "favorite" is depends. In school math they use 10 because of digits and place value (and fingers). In calculus they use e because the derivative of $e^x$ is special (it's $e^x$ again!). In computer science they use 2 because of binary storage. – Matthew Leingang Feb 22 '18 at 21:17
  • This is how I teach the change of base formula for logarithms: By solving an equation like $2^x = 7$ two ways; one way taking the $log_2$, the other base $10$, say. – pjs36 Feb 23 '18 at 01:06
  • Because there is only one answer. If different methods give you different answers you're doing something wrong. – user202729 Feb 23 '18 at 08:28
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    I have a question- did your book specifically say "use logarithm base 10" or did it just say "use log"? If it said "use log" rather than, say, "ln" (natural logarithm) then, it could well have meant the logarithm in general, to any base. – user247327 Feb 23 '18 at 14:13
  • Yes it specifically said, " we will use log base ten of both sides to solve for x" – Y.M.80 Feb 23 '18 at 17:59
  • @gogators there must be something wrong with me then. I only have eight. – Bob says reinstate Monica Feb 24 '18 at 13:18
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    Because logartithms are found by flipping through the pages of large tables, and these tables are usually published only (or predominantly) with $\log_{10}$. What is this "pocket calculator" you kiddies keep talking about? – Hagen von Eitzen Feb 24 '18 at 14:46
  • It's called HiPER and you can download it for free on the Google Play Store for Android smartphones and tablets. Simply a scientific calculator with many functions. – Y.M.80 Feb 24 '18 at 18:45

9 Answers9

81

This works in any base because $\log_b (a^c) = c \log_b(a)$

The practical reason for using base $10$ was a little old fashioned: it allowed the use of tables of logarithms instead of a calculator and reducing these calculations to addition and subtraction. Calculators then provided a $\log_{10}$ function as a carry-over from tables and just called the button log

My 1953 elementary statistical tables have logarithms of numbers from $1$ to $10$ and anti-logarithms of numbers from $0$ to $1$. Since these are logarithms base $10$, I can easily deal with all numbers because $\log(a \times 10^n)=n +\log(a)$

Here I want $\dfrac{\log 7}{\log 2}$.

  • My tables tell me $\log 7\approx 0.8451_6$ (with the ${\,}_6$ helping interpolation) and $\log 2\approx 0.3010_{22}$. So that leaves me with trying to calculate $\dfrac{0.8451}{0.3010}$. I cannot be bothered to do long division, so instead I try to calculate $\text{antilog}\,({\log 0.8451 -\log 0.3010})$.

  • My tables tell me $\log 8.45 \approx 0.9269_5$ so I write $\log 0.8451 \approx \bar{1}.9269$ (with the $\bar{1}$ because I wanted $\log (8.451\times 10^{-1})= -1+\log 8.451$). Similarly it tells me $\log 3.01 \approx 0.4786_{14}$ so I write $\log 0.3010 \approx \bar{1}.4786$

  • I now calculate by hand $\bar{1}.9269 - \bar{1}.4786 = 0.4483$. My tables tell me $\text{antilog}\,0.448 \approx 2.805_7$ and I interpolate the final $3$ using the ${\,}_7$ to give a final approximate answer of $2.807$. Which is what you got with some clever silicon

Henry
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    Your answer was well constructed and has clarified the question asked. Best regards. – Y.M.80 Feb 22 '18 at 08:49
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    I'm a bit too young to be used to log tables... Care to explain where that subscript 6 (and others) comes from? – ilkkachu Feb 22 '18 at 15:31
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    @ilkkachu: $\log 7.00 \approx 0.8451_6$ and the next number to appear would be $\log 7.01$, so if I wanted say $\log 7.004$ I would add $4 \times 6 / 10 \approx 2$ to the final digit to give about $0.8453$. Similarly if I wanted $\log 2.004$ I would add $4 \times 22 / 10 \approx 9$ to the final digit of $\log 2.00 \approx 0.3010_{22}$ to give about $0.3019$ – Henry Feb 22 '18 at 15:57
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    It's worth mentioning that most programming languages do not define log as base-10 but as the much more sensible base-$e$, which calculators normally call ln. – leftaroundabout Feb 22 '18 at 16:19
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    And then if you put those log tables on sticks, and put the sticks in a holder to help line them up, you have a slide rule. – Acccumulation Feb 22 '18 at 21:52
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    Also, loga x = ( log x ) / ( log a ) = ( ln x ) / ( ln a ) - it's a property. And to extend @leftroundabout's answer, ln is shorthand for log-natural - in this case 'e' – Fox Feb 23 '18 at 06:51
  • Ahh... this reminds me how to solve OP's question without calculator and instead used log table like this... – Andrew T. Feb 23 '18 at 11:20
  • So the "old-fashioned" reason is that log tables are more convenient if they use a single base, and the choice of 10 as that base is an arbitrary choice? – Chaim Feb 23 '18 at 18:37
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    @Chaim No, the choice of 10 as a base is not at all an arbitrary choice for log tables; it's based on the fact that we usually have (and want) numbers in decimal. For example, if we wanted to multiply $325.7$ and $54.62$, then we could write them as $10^2\cdot3.257$ and $10^1\cdot5.462$, then use base-10 log tables to find the answer as (roughly) $10^3\cdot10^1\cdot1.77897=17789.7$. Instead if we had base-$e$ log tables, we'd have to rewrite $325.7$ as $e^5\cdot2.1945$ and $54.62$ as $e^4\cdot1.0004$, use the ln-tables to get an answer of $e^9\cdot2.1954$… then what? Not easy to do by hand :-) – ShreevatsaR Feb 23 '18 at 20:45
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    Good. This is the kind of answer I’d have given. And I’m old enough to have used those log tables in high school. – Lubin Feb 24 '18 at 05:37
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Question 1) As you see, both method arrive at the same answer, however $\log$ base 10 is a natural choice, as stated in the comment by Matti P above.

Question 2) It is because for $a>0,a\neq 1,b>0, c>0, c\neq 1$,$$\log_a b=\frac{\log_c b}{\log_c a}$$In particular in your example, $c=10$, so both method arrives the same answer. The formula is known as the change of base formula, see here.

Tony Ma
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    I was not aware of the change of base formula. Your answer has clarified things for me greatly! – Y.M.80 Feb 22 '18 at 08:51
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    In fact, because of the change of base formula, you only need the one $\log$ function, which can be thought of as natural log (for analysis) or "just log, don't ask questions" (for complexity theory). You could also decide that it's base 10 log (for engineering) or base 2 log (for computer science), but that's uglier. – Kevin Feb 24 '18 at 04:25
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Logarithmic functions enjoy many properties.

One of the very interesting properties of logarithms is the formula called change of base formula.

$$ \log_a x = \frac {\log_b x}{\log_b a} $$

For example $$ \log_2 7 = \frac {\log_{10} 7}{\log_{10} 2} $$

This formula makes finding logarithms in an arbitrary base possible by using only logarithms base $10$ or natural logarithms.

polfosol
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Mohammad Riazi-Kermani
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As others have said, the formula works for a logarithm of any base, because of the change-of-base formula. However, the accepted answer says, “The practical reason for using base 10 was a little old fashioned: it allowed the use of tables of logarithms instead of a calculator and reducing these calculations to addition and subtraction.” This raises the question, why did those tables use base 10? After all, we just explained that the same tricks work for any base at all. If you just needed to pick one base to compile a table of and print as a book, or mark on a slide rule, wouldn’t the least-arbitrary choice have been e?

The answer behind the answer is that base-10 logarithms are the easiest for humans without calculators and used to decimal numbers to work with. The log of 1 is 0, the log of 10 is 1, the log of 100 is 2, and so on up, so any one-digit number has a log of zero point something, any two-digit number has a log of one point something, and so forth. So, without a calculator, what’s the log of 300? Well, log 300 is log 3·100, which is log 3 + log 100. The square root of 10 is a little more than 3, so log 3 is a little less than 0.5, and log 100 is exactly 2, so intuitively it’s a little less than 2.5. That makes it really easy to find the log of any number in scientific notation, or move a decimal point left or right.

If you did a lot of these, which engineers once had to, you’d quickly get an intuitive sense for it. And doing these problems with base 10 lets you use that number sense to catch silly mistakes: “No, that can’t be right, it’s something thousand, so the log has to be three point something.”

Davislor
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8

The use of Log in base 10 is a common choice and we often find it in calculators since our standard system of counting is in base 10 (i.e. number of fingers).

To handle algebraic problems any other choice is fine depending upon the specific problem to solve by logarithmic identities.

For the application in calculus the best choice is the natural logarithm in base e.

user
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Other answers mention the change of base $$ \log_a x = \frac {\log_b x}{\log_b a}. $$ An immediate consequence is that a quotient of logarithms does not depend on the base: $$\frac {\log_ax}{\log_ay}=\frac {\frac {\log_bx}{\log_ba}}{\frac {\log_by}{\log_ba}} =\frac {\log_bx}{\log_by}. $$

Martin Argerami
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The only reason we use base ten for anything (including decimal numbers) is that biological evolution endowed humans with ten fingers. The Mayans are supposed to have used base twenty, apparently because they were barefoot and thought that toes were convenient for counting too.

Mark Miller
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2

Another common type of question in logarithms is

$$2^{x+2} = 3^{2x}$$

Now suppose you have learnt always to use $\log_2$ if the base is 2, and $\log_3$ if the base is 3. You are now stuck! Which base should you use?

On the other hand suppose you have learned always to use $\log_{10}$, then you get $$(x+2)\log2 = 2x\log3$$ which you can solve. It is important that you know the $\log(a^b) = b\log(a)$ rule, and solving using base 10 helps you to practice this method.

So a student who learns only one technique for solving equations with logarithms can learn to use base 10. Better students can then learn to use different bases, if another base is more convenient. This is why the textbook may teach you to use base 10: It is more general and practices an important formula.

As for why "10", as other answers have noted, any base is possible, but base 10 is convenient since decimal numbers are based around 10.

James K
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    That should read "base 10 was convenient". My tables only date from the late 70s, but I did learn to perform multiplication by converting ab into antilog(log(a)+log(b)) (when a slide-rule didn't provide enough accuracy). Using base 10 meant that a single table entry could serve for 3.14, 0.314, and 31.4 and the user just had to adjust the bit before the decimal appropriately. Now that we no longer do that, there is no advantage to base 10 – Martin Bonner supports Monica Feb 22 '18 at 19:57
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    base 10 is still convenient since 10^n is so easy to calculate – James K Feb 22 '18 at 21:06
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Why would the textbook suggest to use log base ten rather than simply using log base two?

Hmmm, if we embrace binary, and use log2(), how about using base 2 for the constants and representation also?

Then $2^x = 7$ becomes $10^x = 111$ and is solved with $x = 10.1100111010...$

When numbers are represented in base 10. Using logn() other than $n==10$ takes an unnecessary additional manipulation and prevents observing niceties like log2(100) == 2.

How does base ten play a factor in the whole scope of things. (?)

As base 10 is common understood by the target audience (students), using log10() is also more readily understood by them. It is more instructive than employing base 2.

On other hand, if the textbook was for processors, using base 2 makes more sense.

chux - Reinstate Monica
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