Logarithms & Logarithmic Functions

Q: What's the difference between $\log x$ and $\ln x$?

Only the base. $\log x$ (no base shown) is base $10$, and $\ln x$ is base $e \approx 2.718$. Both ask the same kind of question — what exponent on the base gives $x$ — they just use different bases.

A logarithm answers one question: what exponent do I need? When you see $\log_2 8$ , it's asking — $2$ raised to what power gives $8$ ? Since $2^3 = 8$ , the answer is $3$ . That's it. A logarithm is not a new operation you have to build from scratch; it's the exponent itself, pulled out into the open.

Students who struggle with logs are almost always struggling with the notation, not the idea. You already know that $2^3 = 8$ . The statement $\log_2 8 = 3$ says the exact same thing, just rearranged so the exponent is the answer. Learn to translate between the two forms and most log problems collapse into exponent problems you can already do.

A logarithm asks: what exponent?

The expression $\log_b x$ reads "log base $b$ of $x$ ," and it stands for the exponent you'd put on $b$ to produce $x$ . So $\log_3 81 = 4$ because $3^4 = 81$ , and $\log_{10} 1000 = 3$ because $10^3 = 1000$ .

The logarithm is the inverse of the exponential with the same base. Exponentials take an exponent and give you the result: $2^6 = 64$ . Logarithms take the result and give you back the exponent: $\log_2 64 = 6$ . Same relationship between the numbers, read in the opposite direction.

This is why logs matter beyond the classroom. A bacteria population that doubles every hour follows $2^t$ — an exponential. If you're asked how many hours it takes to reach $64$ cells, you're asking for the exponent, so the answer is a logarithm: $\log_2 64 = 6$ hours.

Switching between log form and exponential form

The two forms are $\log_b x = y$ and $b^y = x$ . Same base $b$ , same statement. The number sitting as the subscript in log form becomes the base of the power in exponential form, and the answer to the log is the exponent.

When a log problem stumps you, rewrite it in exponential form immediately. "Evaluate $\log_5 125$ " becomes "solve $5^y = 125$ " — and since $125 = 5^3$ , you can read off $y = 3$ . Nearly every intro log problem is one rewrite away from being an exponent problem.

One restriction to know: the base $b$ must be positive (and not $1$ ), and you can only take the log of a positive number. There's no real answer to $\log_2(-8)$ , because $2$ raised to any real power is always positive — it can never produce a negative number.

Special values worth memorizing

$\log_b 1 = 0$ for every base, because $b^0 = 1$ . And $\log_b b = 1$ , because $b^1 = b$ . These two facts anchor the whole log scale — every log curve passes through the point $(1, 0)$ .

Two bases show up so often they get shorthand. A log written with no base, like $\log 100$ , means base $10$ — the common log — so $\log 100 = 2$ . The natural log $\ln x$ means base $e \approx 2.718$ ; it's the version calculus and science use constantly. Both work exactly like every other log: they ask for an exponent.

Worked examples

Example 1: evaluate a basic log

Evaluate $\log_2 8$ .

Translate the question:

2

to what power is

8

2^y = 8

Write

8

as a power of

2

8 = 2^3

The exponent is the answer

\log_2 8 = 3

Answer: $\log_2 8 = 3$

Example 2: a bigger base

Evaluate $\log_5 125$ .

Rewrite in exponential form

5^y = 125

Write

125

as a power of

5

125 = 5^3

Read off the exponent

\log_5 125 = 3

Check:

5^3 = 5 \cdot 5 \cdot 5 = 125

✓

Answer: $\log_5 125 = 3$

Example 3: a fraction inside the log

Evaluate $\log_3 \dfrac{1}{9}$ .

Rewrite in exponential form

3^y = \dfrac{1}{9}

A fraction like this needs a negative exponent

\dfrac{1}{9} = \dfrac{1}{3^2} = 3^{-2}

Read off the exponent

\log_3 \dfrac{1}{9} = -2

Answer: $\log_3 \dfrac{1}{9} = -2$

Example 4: a common log (base 10)

Evaluate $\log 10{,}000$ .

No base written means base

10

\log_{10} 10{,}000

Write

10{,}000

as a power of

10

10{,}000 = 10^4

Read off the exponent

\log 10{,}000 = 4

Answer: $\log 10{,}000 = 4$

Try one yourself

Common questions

Is a logarithm just an exponent?

Yes — that's the cleanest way to say it. $\log_b x$ is the exponent you put on $b$ to get $x$ . The statement $\log_2 32 = 5$ and the statement $2^5 = 32$ contain identical information.

Why can't I take the log of a negative number or zero?

Because a positive base raised to any real power stays positive. $2^y$ can be huge or tiny, but it's never zero and never negative — so there's no real exponent that answers $\log_2(-8)$ or $\log_2 0$ .

What's the difference between

\log x

and

\ln x

Only the base. $\log x$ (no base shown) is base $10$ , and $\ln x$ is base $e \approx 2.718$ . Both ask the same kind of question — what exponent on the base gives $x$ — they just use different bases.

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Logarithms & Logarithmic Functions

A logarithm asks: what exponent?

Switching between log form and exponential form

Special values worth memorizing

Worked examples

Example 1: evaluate a basic log

Example 2: a bigger base

Example 3: a fraction inside the log

Example 4: a common log (base 10)

Try one yourself

Common questions

Keep learning

Properties of Logarithms

Solving Exponential Equations & Inequalities

Product & Quotient Rules