If a is a positive real number other than 1, and aᵐ = x, then m is called what?

The logarithm of x to the base a. Here a^m = x, with a positive and a not equal to 1. This is the defining relation for logarithms. By definition, m is written log_a(x). So m is called the logarithm of x to the base a. It is simply the exponent, rewritten as a log statement.

The product rule of logarithms states that log_a(xy) equals what?

log_a(x) + log_a(y). The product rule states log_a(xy) = log_a(x) + log_a(y). Here x and y are any positive numbers, with base a positive and a not equal to 1. This law turns a log of a product into a sum of logs. For example, log_a(x) + log_a(y) recombines into log_a(xy), and the reverse also holds.

What is the value of log_a(1), for any valid base a?

0. Here we need log_a(1), for any valid base a (a > 0, a not equal to 1). Convert to exponential form: log_a(1) = m means a^m = 1. Any nonzero number raised to the power 0 equals 1. So a^0 = 1 for every valid base a. That means m = 0. So log_a(1) = 0. Check: a^0 = 1, confirming the result.

When a logarithm's base is not mentioned, it is conventionally taken to be what?

10. When a logarithm is written without a base, like log(x), the base is taken as 10. This is a fixed convention. A base-10 logarithm is called the common logarithm. So log(x) means log₁₀(x), unless another base is stated. A base-e logarithm is instead written ln(x).

If log base x of 81 = 2, what is the value of x?

9. Here log base x of 81 = 2. Converting to exponential form: log_x(a) = k means x^2 = a. So x^2 = 81. To solve for x, take the square root of both sides. So x = 81^(1/2) = 9. Check: 9^2 = 81.

What is the value of (log base 4 of 64) + (log base 4 of 16)?

5. Here we need (log base 4 of 64) + (log base 4 of 16). The law log_a(x) + log_a(y) = log_a(xy) applies here. So the sum equals log base 4 of (64 x 16). Now 64 x 16 = 1024. Since 64 = 4^3 and 16 = 4^2, we get 1024 = 4^(3+2) = 4^5. By definition, log base 4 of 4^5 = 5. So the answer is 5.

What is the value of log base 3 of (81)^4?

16. Here we need log base 3 of (81)^4. The power rule log_a(x^p) = p x log_a(x) applies here. First find log base 3 of 81. Since 81 = 3^4, log base 3 of 81 = 4. Now apply the power rule: 4 x 4 = 16. So log base 3 of (81)^4 = 16. Check: 3^16 = (3^4)^4 = 81^4 = 43046721.

If log base 25 of p = 3 and log base 5 of q = 3, what is the relation between p and q?

p^1 = q^2. Here log base 25 of p = 3, so p = 25^3. Since 25 = 5^2, the law (a^m)^n = a^(mn) gives p = (5^2)^3 = 5^6. Also log base 5 of q = 3, so q = 5^3 = 5^3. Both p and q are powers of 5, with exponents 6 and 3. To relate them, raise each to make the total exponent match: p^1 has exponent 6 x 1 = 6. q^2 has exponent 3 x 2 = 6. These are equal, so p^1 = q^2.

Logarithms: Formulas, Shortcuts and Solved Examples

A logarithm is just a power in disguise.

This lesson gives each rule with the reason it works. It solves five examples step by step and lists the traps that cost marks.

The idea in brief

Definition: aᵐ = x means log_a x = m.
Product: log xy = log x + log y.
Quotient: log x/y = log x − log y.
Power: log xᵖ = p log x.
Special: log 1 = 0, log of base = 1.

What a Logarithm Is

What is a logarithm?

A logarithm answers one question. What power of the base gives this number?

If 2 to the power 5 is 32, then the log of 32 to base 2 is 5.

We write this as log₂ 32 = 5.

What is the exact definition?

If a is positive and not 1, and aᵐ = x, then m is the logarithm of x to the base a.

Why the base is not 1: every power of 1 is 1, so no other number could ever be reached.

Powers and logarithms are two ways of writing the same fact. 2⁵ = 32 is the same as log₂ 32 = 5.

What if no base is written?

The base is taken as 10. These are called common logarithms.

So log 1000 = 3, because 10 to the power 3 is 1000.

Why do we need logarithms?

Logs turn hard multiplication into easy addition. Before calculators, people used log tables for this.

Today logs are used for sound loudness, earthquake size and the pH scale.

In exams, they test how well you handle powers.

Check 1

If aᵐ = x, then m is the logarithm of x to the base a.

Core Rules and Why They Work

What is the product rule?

log(xy) = log x + log y

Why it works: multiplying numbers adds their powers. 2³ × 2² = 2⁵, so 3 + 2 = 5.

What is the quotient rule?

Divide the numbers, and you subtract the logs.

Why it works: dividing numbers subtracts their powers. 2⁵ ÷ 2² = 2³.

What is the power rule?

log(xᵖ) = p × log x

Why it works: raising a power to a power multiplies the powers. (2²)³ = 2⁶, so 3 × 2 = 6.

Each doubling of the number adds exactly 1 to its base-2 logarithm. This is why logs turn multiplying into adding.

Which two values must I remember?

log of 1 = 0 and log of the base itself = 1.

Why they work: any base to the power 0 is 1. Any base to the power 1 is itself.

What is the change-of-base formula?

log_a x = log_b x ÷ log_b a, for any valid base b.

Why it works: it lets you move every log to a base you know, such as 10.

What are the characteristic and the mantissa?

The whole-number part of a common log is the characteristic.

The decimal part is the mantissa.

For a number above 1, the characteristic is one less than the digits before the decimal point.

So 4567 has 4 digits, and its characteristic is 3.

Which powers should I know by heart?

Powers of 2: 2, 4, 8, 16, 32, 64, 128, 256.
Powers of 3: 3, 9, 27, 81, 243.
Powers of 5: 5, 25, 125, 625, 3125.
Powers of 7: 7, 49, 343, 2401, 16807.

With these, most log questions take under a minute.

Can a log be negative or a fraction?

Yes. If the number is below 1, the log is negative.

For example, log base 2 of 1/4 is −2, because 2 to the power −2 is 1/4.

If the number lies between two powers, the log is a fraction or decimal. For example, log base 2 of 6 lies between 2 and 3.

Check 2

The logarithm of 1 to any valid base is 0.

Five Solved Examples

Easy, step by step: if log base x of 81 = 2, find x.

Step 1: change to a power. x² = 81.

Step 2: take the square root. x = 9.

Step 3: the base must be positive, so reject −9.

The answer is 9.

Easy, step by step: find log base 4 of 64 + log base 4 of 16.

Step 1: use the product rule. This is log base 4 of (64 × 16).

Step 2: 64 = 4³ and 16 = 4², so the product is 4⁵.

Step 3: the log of 4⁵ to base 4 is 5.

The answer is 5.

Check by parts: 3 + 2 = 5.

Medium, step by step: find log base 3 of 81⁴.

Step 1: use the power rule. The value is 4 × log base 3 of 81.

Step 2: 81 = 3⁴, so log base 3 of 81 = 4.

Step 3: 4 × 4 = 16.

The answer is 16.

Medium, step by step: log base 25 of p = 3 and log base 5 of q = 3. Relate p and q.

Step 1: p = 25³. Since 25 = 5², p = (5²)³ = 5⁶.

Step 2: q = 5³.

Step 3: p = 5⁶ = (5³)². So p = q².

The answer is p = q².

Exam level, step by step: find log base 2 of 32², then log base 5 of 25 + log base 5 of 25.

Step 1: log base 2 of 32² = 2 × 5 = 10, since 32 = 2⁵.

Step 2: log base 5 of 25 = 2, so the second sum is 2 + 2 = 4.

Step 3: both can be checked by turning them into powers.

The answers are 10 and 4.

Check 3

log(x + y) is the same as log x + log y.

Common Traps and Shortcuts

Which traps cost marks?

Writing log(x + y) = log x + log y. The rule works for products only.

Forgetting that the base must be positive and not 1.

Taking the log of zero or a negative number. Neither has a log.

Mixing up the product rule and the power rule.

Forgetting that “log” with no base means base 10.

Which shortcuts save time?

Write each number as a power of the base before you start.

Learn powers of 2, 3, 5 and 7 up to the fifth or sixth.

For “log base x of N = k”, just solve x to the power k = N.

For a sum of logs with the same base, multiply the numbers first.

What patterns come up in exams?

Five patterns repeat. Find the base, add two logs, find the log of a power, relate p and q, and recall a rule.

Each one needs one rule and a table of powers.

Check 4

The logarithm of the base itself is always 1.

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