The laws of logarithms

Higher Level

With x = 4 fixed, the green curve is log 4 + log y and the red one is log(4 + y). Slide y and read the gap.

y = 25

2.000log 4 + log y

1.462log(4 + y)

0.538apart

They are never equal, and they are not close.

The three laws

log_a(xy) = log_ax + log_ay

log_a(xy) = log_ax − log_ay

log_a(xⁿ) = n log_ax

Read them one way round and they make sense. A logarithm is a power, powers add when you multiply, so logs add when you multiply. There is no law for log(x + y) because adding powers is not what addition does.

log 4 + log 25= log 100 = 2 log(4 + 25)= log 29 = 1.462 gap= 0.538, not a rounding difference log(2¹⁰)= 10 log 2 = 3.010 ln(e³)= 3

In examinations the base will be 10 or e, so every law above is available on the calculator. The marks are for combining them into a single log before solving, not for the arithmetic.

What they are for

The reason to learn them is to get an unknown out of an exponent.

given 3^x = 50 take logs→ x log 3 = log 50 so= x = log 50log 3 = 3.56

The third law is what moved x from the exponent to the front, where you can divide by its coefficient. That single step is why the law exists.

Worked example

A cup of coffee left in an air-conditioned Bangkok classroom cools so that its temperature above room temperature is T = 65 × 0.94^t, with t in minutes.

How long until it has cooled to 20 degrees above room temperature? Give it to the nearest minute.

65 × 0.94^t = 20Set the model equal to the value asked for.
0.94^t = 0.3077Divide by 65 first, so the power stands alone.
t log 0.94 = log 0.3077The third law: a power comes down to the front, which is the only reason the law is worth knowing.
t = log 0.3077 ÷ log 0.94Both logs are negative, so the quotient is positive. A negative time would mean a slip.
= 19.0, so 19 minutesCheck it: 65 × 0.94¹⁹ = 20.1, which is the rounding at work, not an error. And note that dividing two negative logs gave a positive answer, which is the step students most often panic about.

Your turn

1. Find log 4 + log 25.

2. Write 2 log x − log y as a single logarithm. Enter 1 for log(x²/y), 2 for log(2x/y), 3 for log(2x − y).

3. log(x + y) equals:

log x + log y log x × log y Nothing simpler; there is no such law. log x − log y

Where the marks go

Combining into a single logarithm before solving. Two logs on one side is almost never the finished line.

The coefficient going inside as a power, so 2 log x becomes log x² and not log 2x.

Checking the answer is valid: a log needs a positive argument, so a solution that makes one negative is rejected, and saying so earns the mark.

Come back to this

These three are deliberately not all about this page. In an examination the hardest step is deciding which method applies, and questions that arrive straight after the method never make you decide. Answer from memory before you reveal anything.

Come back to these in a week, and again in a month.

Which method?

Simplify 5³ × 5⁻⁶.

Index law, not a log law. Add the powers: 5⁻³ = 1/125. A negative power is a reciprocal.
Which method?

A loan's first payment is 1288.60 on a 200000 debt at 6% a year, monthly. How much comes off the debt?

Amortization: interest is 0.5% of 200000 = 1000, so 288.60 comes off. The payment is mostly interest at the start.
Which method?

Write 2 log x + log y as a single logarithm.

Log laws: the coefficient becomes a power, then adding multiplies: log(x²y). Not log(2xy).

The same idea elsewhere

1.5, exponents and logarithms: The laws are consequences of the index laws, which is why they look alike.
4.9, the normal distribution: Log scales turn multiplicative data additive, which is how skewed data gets modelled.

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