Change of Base and Interpreting Log Scales

1) Why a “change of base” formula should exist

A logarithm answers an exponent question: log_b(x) is the number y such that b^y = x. The key idea behind change of base is that “being an exponent” does not depend on which base you use to measure it; the base only changes the unit you’re measuring in.

To see why, suppose y = log_b(x). Then b^y = x. Now take a logarithm of both sides using some other base a (any valid base, typically 10 or e):

• Start with b^y = x
• Apply log_a to both sides: log_a(b^y) = log_a(x)
• Use the power rule: y · log_a(b) = log_a(x)
• Solve for y: y = log_a(x) / log_a(b)

Since y = log_b(x), we get the change-of-base relationship:

log_b(x) = log_a(x) / log_a(b)

This is not a calculator trick; it is a statement that the same exponent can be expressed in different “units” (different bases) by scaling.

Two useful special cases

• Using common logs (base 10): log_b(x) = log(x) / log(b)
• Using natural logs (base e): log_b(x) = ln(x) / ln(b)

2) Computing and comparing logs via change of base (meaning-first)

When you compute log_b(x) using ln or log, you are doing a ratio: “how many base-a exponent units does x represent” divided by “how many base-a exponent units does one factor of b represent.”

Continue in our app.

• Listen to the audio with the screen off.
• Earn a certificate upon completion.
• Over 5000 courses for you to explore!

Or continue reading below...

Download the app

Example A: Compute log_2(8) using another base

We already know conceptually that 2^3 = 8, so the answer should be 3. Now express it as a ratio in base 10 (or base e):

• Change base: log_2(8) = log(8) / log(2)
• Interpretation: “How many powers-of-10 steps to reach 8” divided by “how many powers-of-10 steps to double.”

You do not need the decimal values to understand why this must equal 3: since 8 = 2^3, applying the derivation above forces log(8) = 3·log(2), so log(8)/log(2) = 3.

Example B: Compare log_2(10) and log_3(10) without heavy computation

Both ask: “what exponent gives 10?” But the bases grow at different speeds.

• Because 2 is smaller than 3, you need a larger exponent on 2 to reach 10 than you need on 3.
• So log_2(10) > log_3(10).

Change of base supports this comparison cleanly:

log_2(10) = ln(10)/ln(2)   and   log_3(10) = ln(10)/ln(3)

Since ln(10) is the same positive numerator, the larger value comes from the smaller denominator. Because ln(2) < ln(3), we get ln(10)/ln(2) > ln(10)/ln(3).

Example C: Compute a non-integer log as “how many multiplications”

Consider log_10(2). This is the exponent y such that 10^y = 2. It measures “how far you go on a base-10 exponent scale to multiply by 2.” Using natural logs:

log_10(2) = ln(2)/ln(10)

Meaning: the “exponent distance” from 1 to 2, measured in base e, divided by the “exponent distance” from 1 to 10, measured in base e. It is a fraction because multiplying by 2 is less than multiplying by 10.

3) What changing the base really changes: units of “exponent measure”

Think of log_b(x) as a measurement of multiplicative size using base b as the unit step. One step on a base-b log scale means “multiply by b.”

• On a base-2 log scale, +1 means “double.”
• On a base-10 log scale, +1 means “multiply by 10.”
• On a base-e log scale, +1 means “multiply by e.”

So changing base is like changing units (inches to centimeters), except the “quantity” is multiplicative growth. The change-of-base ratio tells you the conversion factor between these units:

log_b(x) = log_a(x) / log_a(b)

Read it as: “value in base-b units” = “value in base-a units” divided by “how many base-a units make one base-b unit.”

Growth-factor comparison through base changes

Suppose one process multiplies by 1.5 each step and another multiplies by 1.2 each step. If you want to compare them on a common “exponent ruler,” you can measure both using the same base (often e or 10):

• Measure “per-step multiplicative size” as ln(1.5) versus ln(1.2).
• Or measure in base 10 as log(1.5) versus log(1.2).

The base doesn’t change which factor is larger; it changes the numerical scale used to report it. Any base change is a constant rescaling.

4) Logarithmic scales: compressing multiplicative ranges

A logarithmic scale is used when values span large multiplicative ranges. Instead of equal spacing representing equal differences, equal spacing represents equal ratios.

Equal steps mean equal ratios

On a base-10 log scale:

• Moving from 1 to 10 is +1 (a factor of 10).
• Moving from 10 to 100 is also +1 (another factor of 10).
• Moving from 100 to 1000 is also +1 (another factor of 10).

So the numbers 1, 10, 100, 1000 are equally spaced on a base-10 log axis even though their differences (9, 90, 900) are not equal. The scale is designed to make multiplicative change look additive.

Step-by-step: reading ratios from log differences

If a log scale uses base b, then a difference of Δ on that log scale corresponds to a multiplicative factor of b^Δ in the original quantity.

• Start with two positive values x1 and x2.
• Compute the log difference: log_b(x2) - log_b(x1).
• Interpretation: this difference equals log_b(x2/x1), so it measures the ratio x2/x1 in “base-b steps.”

In particular:

• If log_b(x2) - log_b(x1) = 1, then x2/x1 = b.
• If the difference is 2, then x2/x1 = b^2.
• If the difference is 0.5, then x2/x1 = b^{0.5} (a square-root factor).

Changing the base of a log scale changes the tick labels, not the structure

Because of change of base, any log scale can be re-expressed in another base by a constant rescaling. If you have a base-10 log value and want the base-2 log value, you multiply by a constant:

log_2(x) = log_10(x) / log_10(2)

This means both scales preserve the same ordering and the same ratio information; they just report distances in different units (doublings versus tenfolds).

5) Applied interpretations: “orders of magnitude” and other log-scale statements

Orders of magnitude (base 10 thinking)

An “order of magnitude” usually means a factor of 10. On a base-10 log scale, that is a change of 1.

• “Two orders of magnitude larger” means about 10^2 = 100 times larger.
• “Half an order of magnitude” means a factor of 10^{0.5} = √10 (about 3.16) times larger.

So if quantity A is 2 orders of magnitude larger than quantity B, then:

log_10(A) - log_10(B) = 2  ⇔  A/B = 100

Decibels-style statements (log differences as ratios, no domain needed)

Many “log units” are built from a constant times a base-10 logarithm of a ratio. Even without knowing the specific field, you can interpret the structure:

Score = k · log_10(x2/x1)

• If x2/x1 multiplies by 10, the score increases by k.
• If x2/x1 multiplies by 100, the score increases by 2k.
• If x2/x1 is 1 (no change), the score is 0.

The constant k just sets the size of one “unit” on that reporting scale.

Interpreting “log-linear” plots in plain language

If a graph uses a logarithmic scale on the vertical axis, then equal vertical steps correspond to equal multiplicative changes in the underlying quantity. This lets you interpret patterns quickly:

• A straight line on a log-scaled vertical axis indicates a constant multiplicative factor over equal horizontal steps (because equal log increases mean equal ratios).
• Vertical gaps are best read as ratios, not differences. For example, a gap of 1 on a base-10 axis means “10 times,” regardless of where you are on the axis.

Quick translation table: log differences to factors

The main habit to build is this: on a log scale, differences represent ratios, and changing the base changes only the unit used to report those ratios.