A logarithm asks an exponent question backwards. How many times must 2 be multiplied to make 8? The answer is 3, written log₂ 8 = 3.
So 2³ = 8 and log₂ 8 = 3 state the same fact from two directions.
Logarithms turn multiplication into addition, which makes quantities that differ by whole factors, like earthquake magnitudes or sound in decibels, much easier to handle.
Change the base a, and both the steepness of the exponential curve in the first picture and the gap between the two scales in the second change with it.
Exponents: multiplying the same number again and again
When the base and exponent are known, as in 2⁵ = 32, multiplication gives the value at once. Turn it around, give only the base 2 and the value 32, and ask for the exponent, and multiplication has no symbol for the answer; the name for that blank is the logarithm. logₐ N means the power to which a must be raised to give N, and it needs a base with a > 0 and a ≠ 1 and a number N > 0.
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The curve y = aˣ for the chosen base, with dots at whole-number x that grow a times per step
The gold curve in the first picture is y = aˣ, with dots at x = 0, 1, 2, 3, 4, 5. Each step from one dot to the next multiplies the height by a, so the curve hugs the floor at first and shoots up at the right end. The top of the vertical axis is a⁵, only about 2.49 when the base is 1.2 but 1024 when it is 4, and the left-hand dots get squashed down near the dotted line at height 1 accordingly.
2
💡 An exponent counts multiplications
①2³ = 2 × 2 × 2 = 8 is 2 multiplied three times
②Each time x grows by 1, aˣ is multiplied by a, so a bigger base grows faster
③If a > 1 the values grow, and if 0 < a < 1 they shrink
④The base in this picture is at least 1.2, so only the growing case is drawn
Logarithms: asking how many times
In logₐ N the small number written low is the base and the number after it is the argument. Swap them to log₈ 2 and the question becomes how many times 8 must be multiplied to give 2, whose answer is 1/3, not 3. Another common slip is splitting log₂(8 + 4) into log₂ 8 + log₂ 4; a log only splits into a sum when its argument is a product. In fact log₂ 12 is about 3.58, while log₂ 8 + log₂ 4 is 5.
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The same values a⁰ to a⁴ marked on a log scale (top) and an ordinary scale (bottom)
💡 A logarithm asks back how many times
①log₂ 8 = 3 means the answer to "how many 2s multiply to 8?" is 3
②On a log scale each step is the same factor, so 1, a, a², a³, a⁴ sit at equal gaps
③Earthquake magnitude, sound in decibels and acidity in pH all use log scales
Laws of Exponents & Logs
Exponent Laws
am × an = am+n, am ÷ an = am-n
Multiplying powers of the same base adds exponents; dividing subtracts them
Definition of Log
ax = N ⟺ x = loga N
The power to which a must be raised to give N
Log Laws
loga MN = loga M + loga N
The log of a product is the sum of the logs (the exponent law, rewritten in logs)
Change of Base
loga b = logc blogc a
Writing top and bottom as logs with the same base c lets you switch to any base
How exponents and logs are related
🔗 Two faces of the same fact
①Exponent: base and exponent give the value (2³ = 8)
②Logarithm: base and value give the exponent (log₂ 8 = 3)
③So either statement can be rewritten as the other at once
④The graphs of y = aˣ and y = logₐ x are mirror images in the line y = x
⑤The common logarithm, with base 10, comes later in this chapter
Core Identities
alog_a N = N, loga ax = x
Exponents and logarithms undo each other
Summary
The heart of exponents and logs
loga N = x ⟺ ax = N
A logarithm is an exponent asked backwards
🎯 Exam Points
①Definition: aˣ = N ⟺ logₐ N = x
②Conditions: base a > 0, a ≠ 1, argument N > 0
③Log laws: products become sums, quotients become differences, powers become coefficients
④Change of base: logₐ b = log b / log a (top and bottom just need the same base)
⑤logₐ(M + N) is not logₐ M + logₐ N
Worked Examples & Past Exam
Example 1
Evaluate log2 8 + log2 4.
1
A sum of logs with the same base becomes the log of a single product.
log2 8 + log2 4 = log2 (8 × 4) = log2 32
2
Since 32 = 2⁵, multiplying 2 five times gives 32.
log2 32 = log2 25 = 5
▸ 5
Because 8 = 2³ and 4 = 2², the answer 5 is just the sum of the exponents 3 + 2. That addition of exponents is exactly why a sum of logs becomes the log of a product.
Example 2
If log2 3 = a, express log2 24 in terms of a.
1
Split 24 into a power of 2 times the leftover factor 3, that is, 2³ × 3.
log2 24 = log2 (23 × 3) = log2 23 + log2 3
2
log₂ 2³ = 3 and log₂ 3 = a.
= 3 + a
▸ a + 3
Splitting the number into a power of the base and a leftover factor turns the power part into a whole number and leaves only the leftover as the given letter.
2023 CSAT Math type, adapted
If log3 5 = a and log3 4 = b, express log3 80 in terms of a and b. (Note: 80 = 16 × 5)
①a + b
②a + 2b
③2a + b
④ab
⑤2a + 2b
▸ ② a + 2b
1
Writing 80 as 4² × 5 lets everything be expressed with the given log₃ 4 and log₃ 5.
log3 80 = log3 (42 × 5) = log3 42 + log3 5
2
The exponent of a power comes down as a coefficient, so log₃ 4² = 2b, and log₃ 5 = a.