Definition of a logarithm

When you reach for it

Some equations put the unknown where none of the usual tools can reach it. In

the x is an exponent. Adding, subtracting, multiplying, dividing, squaring and taking roots all act on the number 2^x as a whole; none of them brings the x down out of the exponent. You can still say a good deal about the answer. Since 2^3 = 8 and 2^4 = 16, and powers of 2 grow as the exponent grows, x lies between 3 and 4. The graph shows there is exactly one such x.

What is missing is a name for that number. It is not a fraction you can write down, and it is not a root, so the notation so far has nothing to call it. The logarithm is that name: \log_2 12 means “the exponent that turns 2 into 12”. With it the equation is solved exactly,

So reach for this rewrite whenever the unknown sits in an exponent, as in an exponential function asked “when does it reach this value?”. Reach for it in the other direction too: a statement about a logarithm is often easiest to understand, or to solve, once it is turned back into the power statement it stands for.

The procedure

The definition

Let b be a positive number other than 1, and let x be a positive number. The logarithm of x with base b, written \log_b x, is the exponent that b must be raised to in order to give x:

The two statements say exactly the same thing in two notations. The left one is the logarithmic form, the right one the exponential form. It is read “log base b of x equals y”. The number x is called the argument of the logarithm.

The three numbers keep their jobs when you switch forms, and that is the whole procedure.

• The base b is the base in both. It sits low beside “log”, and it is the number being raised to a power.
• The logarithm y is an exponent. It stands alone on one side of the logarithmic form and sits up in the exponent in the exponential form.
• The argument x is the result of the power. It follows the “log”, and in the exponential form it is what the power equals.

A logarithm is an exponent. Every time you see \log_b x, you can ask “what power of b gives x?”, and the answer is the logarithm.

Why the definition gives exactly one number

For \log_b x to be one definite number, the equation b^y = x must have exactly one solution y. Two facts about exponential functions guarantee it when b > 0, b \neq 1 and x > 0.

There is at most one y. For b > 1 the powers b^y strictly increase as y increases, and for 0 < b < 1 they strictly decrease. Either way two different exponents never give the same power, so the function y \mapsto b^y is one-to-one.

There is at least one y. The powers b^y are always positive, and as y runs over every real number they take every positive value, with no gaps. This is what the picture above shows for base 2: any horizontal line above the x-axis meets the curve, and meets it once.

Because the exponential is one-to-one, it has an inverse function, and the logarithm with base b is that inverse. Switching between the two forms is the same move as switching between y = f(x) and x = f^{-1}(y) for any inverse pair.

Two special notations

Two bases are used so often that their logarithms have their own names.

• The common logarithm has base 10 and is written without the base: \log x means \log_{10} x.
• The natural logarithm has base e \approx 2.718, the base of the natural exponential function, and is written \ln x. So \ln x = y means e^y = x.

Both follow the definition exactly. Only the way the base is shown changes.

What the definition says straight away

Four facts follow from the definition and the rules for powers, with nothing else needed. For every allowed base b:

• \log_b 1 = 0, because b^0 = 1, as the zero exponent rule says.
• \log_b b = 1, because b^1 = b.
• \log_b \left(b^y\right) = y for every real number y. The exponent that turns b into b^y is y itself.
• b^{\log_b x} = x for every positive x. The number \log_b x is, by definition, the exponent that turns b into x, so raising b to it gives x.

The last two say that the exponential and the logarithm with the same base undo each other, which is what being inverse functions means.

Worked examples

Boundaries

The base must be positive and not 1. For b = 1, every power 1^y equals 1, so \log_1 1 would be every number at once and \log_1 5 would be no number at all; neither is a single value. A base of 0 fails the same way, since 0^y is 0 for every positive y and is undefined otherwise. A negative base fails because its powers are not real numbers for many exponents: \displaystyle (-2)^{\frac{1}{2}} would be a square root of -2. Only with b > 0 and b \neq 1 does every positive argument have exactly one logarithm.

The argument must be positive. Every power of a positive base is positive, so no real exponent turns b into 0 or a negative number. \log_b 0 and \log_b(-4) have no real value. (Among the complex numbers logarithms of negative numbers do exist, but they are outside real algebra.)

The logarithm itself may be any real number. Positive, negative, zero or a fraction: the restriction is on the base and the argument, never on the value. \displaystyle \log_2 \frac{1}{32} = -5 and \displaystyle \log_9 3 = \frac{1}{2} are as good as \log_5 625 = 4.

The rewrite is an equivalence only inside these conditions. When an equation is solved by switching forms, the switch itself never introduces an error, but a value found for a base or argument must still be allowed. A base solution that is negative, zero or 1 is rejected, and a value of x that makes an argument zero or negative is rejected. Such answers are taken up in solving logarithmic equations.

The power must stand alone before you switch. 5 \cdot 2^n = 1000 is not the exponential form of anything; 2^n = 200 is. Isolate the power first.

Where this goes next

With logarithms named, they can be worked out. Evaluating a logarithm asks “what power of the base gives this number?” and answers it for numbers that work out exactly, and for common and natural logarithms on a calculator. The laws of logarithms are the laws of exponents read through this definition: since powers with the same base multiply by adding exponents, logarithms turn products into sums. The graph of a logarithm is the graph of the exponential reflected in the line y = x, exactly as for any inverse function. And when the power cannot be isolated as neatly as 2^n = 200, taking a logarithm of both sides brings the exponent down, while Change of base turns any base into one a calculator has.

Exercises

Write in exponential form

1. \log_2 128 = 7
2. \displaystyle \log_3 \frac{1}{81} = -4
3. \displaystyle \log_{125} 5 = \frac{1}{3}
4. \log 100{,}000 = 5

Write in logarithmic form

5. 3^5 = 243
6. \displaystyle 2^{-4} = \frac{1}{16}
7. \displaystyle 100^{\frac{3}{2}} = 1000
8. 10^{-2} = 0.01
9. e^k = 9

Find the unknown

10. \log_6 x = 3
11. \log_3 x = -2
12. \displaystyle \log_{36} x = \frac{1}{2}
13. \displaystyle \log_8 x = \frac{5}{3}
14. \log_b 64 = 3
15. \log_b 81 = 4
16. \displaystyle \log_b 4 = \frac{2}{5}
17. \log_7(2x + 3) = 2
18. \log_2(x^2 + 7) = 4

When the exponent is unknown

19. Solve 5^x = 40 exactly, say between which two whole numbers x lies, and give x to two decimal places.
20. Solve 3 \cdot 10^x = 750 exactly, and give x to two decimal places.
21. Between which two whole numbers does \log_2 1000 lie? Explain without a calculator.
22. Find the inverse of g(x) = 4^{x+2}, and state its domain.
23. A patch of pond weed covers 2 square meters and triples in area every week, so after t weeks it covers 2 \cdot 3^t square meters. Find, exactly and to two decimal places, the time at which it covers 150 square meters.

Reasoning

24. Explain why \log_1 5 has no value, and why \log_b 1 = 0 for every allowed base b.
25. Explain why \log_2(-8) has no real value while \displaystyle \log_2 \frac{1}{8} does, and find the second.
26. Use the definition to show that b^{\log_b x} = x for every positive x, and that \log_b\left(b^y\right) = y for every real y.
27. Bastien writes \log_{-2} 4 = 2, because (-2)^2 = 4. Explain why the definition does not allow the base -2, even though this one power works.