Math Vale

⚙️ Function Kingdom · Functions

Logarithms

Read a logarithm as the exponent that answers "b to what power gives x", convert between exponential and logarithmic form, evaluate exact logs, and expand or condense with the product, quotient and power laws.

In short

  • A logarithm is an exponent: log_b(x) is the answer to "b to what power gives x?", so bn = x and log_b(x) = n say the same thing.
  • log_b(1) = 0 and log_b(b) = 1 for every base, and log_b(bk) = k is the whole idea in one line.
  • An argument below 1 gives a negative logarithm, and an argument that is a root of the base gives a fractional one.
  • The three laws are the exponent rules backwards: a product inside becomes a sum outside, a quotient becomes a difference, and an exponent comes down in front.
  • There is no law for the logarithm of a sum — log_b(x + y) cannot be split, and log_b(x) + log_b(y) is log_b(xy) instead.
  • Change of base is log_b(x) = ln(x)/ln(b), argument on top and base underneath, which is what lets a two-button calculator reach any base.
  • On a logarithmic scale such as pH, decibels or Richter, a difference between two readings is a ratio in the real world.

The question a logarithm asks

Doctor Kelvin keeps one sentence carved above the door at Exponential Frost, and everything in this craft is that sentence read out loud:

log_b(x) is the answer to "b to what power gives x?"

That is all a logarithm is. A logarithm is an exponent. It is not a new kind of number, it is not a multiplication and it is not a function that shrinks things — it is the missing power in a statement about a power.

So the two forms below are the same fact wearing two coats.

bn = x (exponential form)

log_b(x) = n (logarithmic form)

Read them across and notice what stays put. The base b stays the base in both. The answer of the logarithm is the exponent n. The number x — the thing inside the bracket, called the argument — is what the power has to reach.

25 = 32 and log_2(32) = 5

103 = 1000 and log(1000) = 3

3-2 = 1/9 and log_3(1/9) = -2

Three details are worth naming now, because they are what the exponent rules already told you.

  • The base is always positive and not 1, because 1 to any power is stuck at 1 and could never reach 32.
  • The argument is always positive, because a positive base raised to any power stays positive. There is no exponent that turns 2 into 0 or into -4, so log_2(0) and log_2(-4) have no value.
  • The logarithm itself can be anything: positive, zero, negative, or a fraction.

Two notations get shortened because they are used so often. log(x) with no subscript means base 10 — the common logarithm — and ln(x) means base e, the natural logarithm, where e is about 2.718. So log(1000) = 3 and ln(e3) = 3, and both are the same question in different clothes.

Exact values, and the ones worth knowing cold

To evaluate a logarithm exactly, turn it back into the question and answer the question.

log_3(81) = ? -> 3 to what power gives 81? -> 34 = 81 -> 4

The reliable method is: write the argument as a power of the base. Once both sides show the same base, the exponent is sitting there in plain sight.

Four values come up so often that they stop being calculations.

  • log_b(1) = 0 for every base, because b0 = 1. This is the one students most often get wrong; it feels as if a logarithm of 1 ought to be 1, and it is 0.
  • log_b(b) = 1, because b1 = b. So log(10) = 1 and ln(e) = 1.
  • log_b(1/b) = -1, because a negative exponent flips the base over.
  • log_b(bk) = k for any k at all. This is the whole idea in one line.

Once the argument is smaller than 1, the answer is negative:

log_5(1/25) = -2 because 5-2 = 1/25

log(0.001) = -3 because 10-3 = 0.001

And once the argument is a root of the base, the answer is a fraction:

log_4(2) = 1/2 because 41/2 = 2

log_8(4) = 2/3 because 82/3 = (23)2/3 = 22 = 4

log_16(8) = 3/4 because both 16 and 8 are powers of 2, and 24 raised to 3/4 gives 23

The trick in that last group is always the same: find the number both the base and the argument are powers of. 8 and 4 are both powers of 2, so write 8 = 23 and 4 = 22, and the logarithm is the ratio of the exponents, 2/3.

Three laws, and where they come from

Because a logarithm is an exponent, the three laws are just the exponent rules read backwards. Nothing here is new — it is arithmetic you already do, looked at from the other end.

The product law. bm * bn = bm + n: multiplying powers adds exponents. So multiplying arguments adds logarithms.

log_b(xy) = log_b(x) + log_b(y)

The quotient law. bm / bn = b^(m - n): dividing powers subtracts exponents.

log_b(x/y) = log_b(x) - log_b(y)

The power law. (bm)k = bmk: a power of a power multiplies exponents. So an exponent inside comes down in front.

log_b(xk) = k log_b(x)

Every one of them needs the same base in every term. log_2(x) + log_3(y) condenses into nothing at all.

Reading the laws left to right is expanding; reading them right to left is condensing. Expanding log_2(8x3/y) uses all three at once:

log_2(8x3/y) = log_2(8) + log_2(x3) - log_2(y)

log_2(8x3/y) = 3 + 3log_2(x) - log_2(y)

Notice the two extra habits in there. Anything on the bottom of the fraction is subtracted, and any logarithm of a plain number gets worked out rather than left standing. A root is a fractional power, so sqrt(x) contributes (1/2)log_b(x).

And now the warning that this whole section exists to give. There is no law for the logarithm of a sum.

log_b(x + y) is NOT log_b(x) + log_b(y)

The right-hand side equals log_b(xy), which is a different number entirely. Test it: log(10 + 10) = log(20), about 1.301, while log(10) + log(10) = 2. When the bracket holds a sum, the laws simply have nothing to say, and the honest answer is that it cannot be expanded.

Change of base, and logarithms out in the world

A calculator only offers two logarithm buttons: log (base 10) and ln (base e). Every other base is reached with the change-of-base formula:

log_b(x) = ln(x) / ln(b) = log(x) / log(b)

The argument goes on top, the base goes underneath — that order is the whole formula, and reversing it is the standard slip. Check it on something you know: ln(8)/ln(2) = 2.0794/0.6931 = 3, and log_2(8) really is 3.

log_2(20) = ln(20) / ln(2) = 2.9957 / 0.6931 = 4.322

Before reaching for a calculator, it is worth trapping the answer between two whole numbers. 24 = 16 is below 20 and 25 = 32 is above it, so log_2(20) lies between 4 and 5 — and 4.322 passes that check.

The natural logarithm has one more property that gets used constantly: e and ln undo each other.

ln(e7) = 7 and eln 5 = 5

Common logarithms are how measuring scales cope with quantities that span billions.

  • pH. pH = -log[H+], where [H+] is a hydrogen ion concentration. A concentration of 10-5 mol/L gives log[H+] = -5, so pH = 5. The minus sign in the formula exists purely to turn those negative exponents into friendly positive numbers.
  • Decibels. Loudness is 10 log(I/I0). A sound 108 times the reference intensity measures 10 * 8 = 80 dB. Do not stop at the logarithm — the factor of 10 is part of the formula.
  • The Richter scale. Magnitude is log(A) for wave amplitude A. Two tremors at 6.4 and 3.4 differ by 3 in magnitude, and log(A1) - log(A2) = 3 means A1/A2 = 103, so the first has 1000 times the amplitude. A difference on a log scale is always a ratio in the real world.

That last line is the reason logarithmic scales exist at all: they turn multiplying by a thousand into adding three.

How to type your answer

The bracket notation in every prompt is the notation you should read: log_2(8) is a base-2 logarithm, log(1000) is base 10, and ln(e3) is base e.

Exact values are typed as a whole number or a fraction in lowest terms. log_3(81) is typed as 4, log_5(1/25) is typed as -2, and log_8(4) is typed as 2/3 (4/6 is marked as not simplified). Never type a rounded decimal where an exact value was asked for.

Expanding, and condensing with a general base, are multiple choice. The answer checker can read log and ln but has no way to read a subscript, so anything containing log_2( ) is offered as options to pick from rather than a box to type in. Read all four before choosing: the wrong ones are the classic slips, not random noise.

Condensing with log or ln asks for the argument on its own. The prompt writes the shape for you — "write this as a single logarithm log(A)" — and you type just A, because A is the whole point: it is the expression that ends up inside the bracket. Write multiplication with * or side by side, powers with ^, and roots as sqrt( ):

2log(x) + log(y) - 3log(z) = log(A) -> A is typed as x2*y/z3

(1/2)ln(x) + 3ln(y) - ln(z) = ln(A) -> A is typed as sqrt(x)*y^3/z

Change of base asks for a decimal to 3 decimal places, so log_2(20) is typed as 4.322. When the question instead asks which two whole numbers a logarithm lies between, give the pair smallest first, as (4, 5).

The definition questions — log_2(x) = 5, log_x(49) = 2, log_3(27) = x — all want a single number, whole or fractional. Whichever piece is missing, rewrite the line as a power first and the missing piece is the only thing left to find.

The pH, decibel and Richter questions all come out as whole numbers, and the unit is already written in the prompt, so just the number is enough.

Worked examples

Example 1

Work out log_8(4) exactly.

  1. Read the question the logarithm is asking: 8 to what power gives 4?
  2. Neither number is a power of the other, so find the number they are both powers of: 8 = 23 and 4 = 22.
  3. Let log_8(4) = n. Then 8n = 4, so (23)n = 22, which is 23n = 22.
  4. Same base, so the exponents must match: 3n = 2, giving n = 2/3.
  5. Check it: 82/3 means the cube root of 8 squared, and 22 = 4. So log_8(4) = 2/3.

Example 2

Expand log_2(8x3/y) as far as the logarithm laws allow.

  1. The bracket holds a quotient, so the quotient law splits it first: log_2(8x3) - log_2(y).
  2. The top is a product, so the product law splits that: log_2(8) + log_2(x3) - log_2(y).
  3. The power law brings the exponent 3 down in front of its own term only: log_2(8) + 3log_2(x) - log_2(y).
  4. log_2(8) is a plain number, so work it out: 23 = 8, so log_2(8) = 3.
  5. The expansion is 3 + 3log_2(x) - log_2(y). Note that y keeps its minus sign because it was on the bottom.

Example 3

Write 2log(x) + log(y) - 3log(z) as a single logarithm.

  1. Every term has base 10, so the laws apply. Deal with the coefficients first, using the power law backwards.
  2. 2log(x) = log(x2) and 3log(z) = log(z3), so the expression is log(x2) + log(y) - log(z3).
  3. Added logarithms multiply their arguments: log(x2) + log(y) = log(x2*y).
  4. The subtracted logarithm divides, so it goes on the bottom: log(x2*y) - log(z3) = log(x2*y/z3).
  5. The single logarithm is log(x2*y/z3). When the prompt writes it as log(A), the part you type is the argument on its own: x2*y/z3.

Example 4

A calculator has only log and ln. Use it to find log_3(40) to 3 decimal places, and check the answer is sensible.

  1. Trap it first: 33 = 27 is below 40 and 34 = 81 is above it, so the answer must lie between 3 and 4.
  2. Change of base puts the argument on top and the base underneath: log_3(40) = ln(40) / ln(3).
  3. ln(40) = 3.6889 and ln(3) = 1.0986.
  4. 3.6889 / 1.0986 = 3.3578..., which rounds to 3.358.
  5. That sits between 3 and 4 as predicted, so the fraction was the right way up. Using log instead of ln gives the same number: 1.6021 / 0.4771 = 3.358.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

Write 53 = 125 in logarithmic form. Which statement says the same thing?

  1. log_5(125) = 3
  2. log_3(125) = 5
  3. log_125(5) = 3
  4. log_5(3) = 125

Answer: A. log_5(125) = 3

  1. The power is 53 = 125: base 5, exponent 3, result 125.
  2. log_b(x) = n asks "b to what power gives x?", so the base 5 keeps its place and the exponent 3 is the value of the logarithm.
  3. Both forms: 53 = 125 and log_5(125) = 3.

Problem 2

Difficulty 3 of 5

Work out log(1/100) exactly. Give a whole number or a fraction in lowest terms.

Answer: -2

  1. log(1/100) asks: 10 to what power gives 1/100?
  2. 10-2 = 1/100.
  3. So log(1/100) = -2.

Problem 3

Difficulty 4 of 5

Expand log(x^5/sqrt(y)) as far as the logarithm laws allow. Which one is right?

  1. 5log(x) - 2log(y)
  2. 5log(x) + (1/2)log(y)
  3. 5log(x) / ((1/2)log(y))
  4. 5log(x) - (1/2)log(y)

Answer: D. 5log(x) - (1/2)log(y)

  1. Look at what is inside the bracket: x^5/sqrt(y). This needs the quotient and power laws.
  2. Product to sum, quotient to difference, exponent down in front — and any log of a plain number is worked out.
  3. log(x^5/sqrt(y)) = 5log(x) - (1/2)log(y)

Common mistakes

  • Swapping the base and the argument: writing 25 = 32 as log_32(2) = 5 instead of log_2(32) = 5. The base of the power stays the base of the logarithm.
  • Answering log_b(1) with 1. Any base to the power 0 gives 1, so the logarithm of 1 is 0 in every base.
  • Splitting the log of a sum: writing log(x + y) as log(x) + log(y). That right-hand side is log(xy), and the laws only ever touch products, quotients and powers.
  • Leaving the exponent on the whole logarithm: writing log(x3) as (log x)3 rather than bringing the 3 down in front as 3log(x).
  • Dropping the minus sign on a reciprocal argument: giving log_5(1/25) as 2 when 5-2 = 1/25 makes it -2.
  • Turning change of base upside down: writing log_5(7) as ln(5)/ln(7). The argument belongs on top; test the formula on log_2(8) whenever you are unsure.
  • Reading a square root as a factor of 2: expanding log(sqrt(x)) as 2log(x) instead of (1/2)log(x), because a square root is the power 1/2.
  • Answering a decibel or Richter question with the logarithm itself and forgetting the rest of the formula — the factor of 10 in decibels, or turning the difference of magnitudes back into a power of 10.

What you should be able to do

  • Convert an equation between exponential form and logarithmic form.
  • Evaluate a logarithm exactly, including common and natural logarithms.
  • Expand and condense logarithmic expressions with the product, quotient and power laws.
  • Use the change-of-base formula and estimate a logarithm from a calculator.

Where this fits in the curriculum

Common Core

  • HSF-LE.A.4

    High school — For exponential models, express as a logarithm the solution to ab^(ct) = d where a, c and d are numbers and the base b is 2, 10 or e; evaluate the logarithm using technology.

  • HSF-BF.B.5

    High school — Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

    HSF-BF.B.5 is a (+) standard — beyond the college- and career-ready threshold, i.e. precalculus rather than Algebra II.

SAT

  • Passport to Advanced Math

    Exponential expressions and the relationship between exponents and logarithms.

    The SAT tests exponential structure and almost never a logarithm by name, so this skill goes further than the test does.

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