What Is a Logarithm?
A logarithm answers the question: "What exponent do I need?" Specifically, log_b(x) = y means b^y = x. The base b is the number being raised to a power, x is the result, and y is the exponent (the logarithm itself).
log_b(x) = y means b^y = x. Every logarithm requires b > 0, b ≠ 1, and x > 0. You cannot take the log of zero or a negative number (in real numbers), and the base cannot be 1 because 1^n = 1 for every n, so the exponent would never be unique.For example, log₂(8) = 3 because 2³ = 8. Similarly, log₁₀(1000) = 3 because 10³ = 1000, and log₅(25) = 2 because 5² = 25.
In other words, logarithms and exponents are inverse operations — they undo each other, just like addition and subtraction or multiplication and division. If exponentiation takes a base and an exponent and produces a result, then the logarithm takes the base and the result and tells you the exponent.
Two Special Logarithms
Two logarithms are so common that they get their own notation. The common logarithm uses base 10 and is written simply as log(x) (no base shown). The natural logarithm uses base e ≈ 2.718 and is written as ln(x). The natural log appears constantly in calculus, science, and finance because of the special properties of the number e.
The Key Logarithm Rules
Logarithms follow a set of rules that mirror the exponent rules, and mastering them is essential for simplifying expressions and solving equations.
log_b(MN) = log_b(M) + log_b(N). The log of a product is the sum of the logs. For example, log₂(4 · 8) = log₂(4) + log₂(8) = 2 + 3 = 5, and indeed 4 · 8 = 32 = 2⁵.log_b(M/N) = log_b(M) − log_b(N). The log of a quotient is the difference of the logs. For example, log₁₀(1000/10) = log₁₀(1000) − log₁₀(10) = 3 − 1 = 2.log_b(M^p) = p · log_b(M). An exponent inside a log can be pulled out front as a multiplier. For example, log₃(81) = log₃(3⁴) = 4 · log₃(3) = 4 · 1 = 4.Identity, Zero, and Inverse Rules
| Rule | Formula | Why |
|---|---|---|
| Zero Rule | log_b(1) = 0 | Because b⁰ = 1 for any valid base. |
| Identity Rule | log_b(b) = 1 | Because b¹ = b. |
| Inverse Rule | b^(log_b(x)) = x | Exponentiation and logarithms cancel. |
| Inverse Rule | log_b(b^x) = x | Exponentiation and logarithms cancel. |
These rules work for any base — common log, natural log, or any other base. They are the tools you will use in nearly every logarithm problem.
Change of Base Formula
Most calculators only have buttons for log (base 10) and ln (base e), so what do you do when you need log₅(37)? You use the change of base formula.
log_b(x) = log_c(x) / log_c(b), where c can be any base you want. In practice, you usually change to base 10 or base e.Using common logs: log₅(37) = log(37) / log(5) = 1.5682 / 0.6990 ≈ 2.243.
You get the same answer using natural logs: ln(37) / ln(5) = 3.6109 / 1.6094 ≈ 2.243.
log₅(37) ≈ 2.243The change of base formula also lets you compare logarithms in different bases and is useful for graphing logarithmic functions.
Why It Works
A quick derivation shows why it works: let log_b(x) = y, so b^y = x. Take log_c of both sides: y · log_c(b) = log_c(x), so y = log_c(x) / log_c(b).
log_b(a) = 1 / log_a(b). This follows directly from the change of base formula and is occasionally useful for simplifying expressions where you see reciprocal logarithms.Solving Logarithmic Equations
Logarithmic equations come in several common types. Here are the strategies for each.
Convert to exponential form. log₃(x) = 4 becomes x = 3⁴ = 81. Always check that your answer keeps the argument positive.
x = 81If log_b(A) = log_b(B), then A = B (as long as both arguments are positive).
For example, log₂(3x − 1) = log₂(x + 7) gives 3x − 1 = x + 7, so x = 4.
Check: 3(4) − 1 = 11 > 0 and 4 + 7 = 11 > 0, so the solution is valid.
x = 4Use the product, quotient, or power rules to combine into a single logarithm, then convert. For example, log(x) + log(x − 3) = 1.
Combine: log(x(x − 3)) = 1.
Convert: x(x − 3) = 10¹ = 10.
Expand: x² − 3x − 10 = 0.
Factor: (x − 5)(x + 2) = 0. So x = 5 or x = −2.
But x = −2 makes log(−2) undefined, so the only solution is x = 5.
x = 5When the variable is in the exponent, take the log of both sides. For example, 5^x = 200.
Take ln of both sides: x · ln(5) = ln(200), so x = ln(200)/ln(5) ≈ 3.292.
x ≈ 3.292Always check your solutions by substituting back into the original equation. Logarithmic equations frequently produce extraneous solutions — values that satisfy the algebra but make a logarithm's argument zero or negative.
Graphing Logarithmic Functions
The graph of y = log_b(x) has several distinctive features.
| Feature | Detail | Why |
|---|---|---|
| Key point | (1, 0) | Because log_b(1) = 0 for any base. |
| Key point | (b, 1) | Because log_b(b) = 1. |
| Domain | x > 0 | Only positive inputs are valid. |
| Range | All real numbers | Logs can take any real value. |
| Vertical asymptote | x = 0 | The curve approaches the y-axis but never touches it. |
Increasing vs. Decreasing
When the base is greater than 1, the function is increasing: as x grows, y grows (but more and more slowly). When the base is between 0 and 1, the function is decreasing.
The graph of y = log_b(x) is the reflection of y = b^x across the line y = x, which visually confirms that logs and exponentials are inverses.
Transformations
Transformations work the same way as with other functions.
| Transformation | Effect |
|---|---|
y = log_b(x − h) + k | Shifts the graph right by h and up by k. |
y = −log_b(x) | Reflects it across the x-axis. |
y = log_b(−x) | Reflects it across the y-axis (and the domain becomes x < 0). |
Understanding the shape of log graphs helps you estimate answers, check reasonableness, and solve inequalities graphically.
Common Mistakes and Tips
Thinking log(a + b) = log(a) + log(b). There is no sum rule for logarithms. The product rule says log(ab) = log(a) + log(b), but that's for multiplication inside the log, not addition. log(2 + 3) = log(5) ≈ 0.699, but log(2) + log(3) = log(6) ≈ 0.778.
Forgetting to check for extraneous solutions. After solving a logarithmic equation, you must verify that every solution keeps all logarithmic arguments positive. Skipping this step is the single most common source of wrong answers.
Confusing log(x²) with (log(x))². The first is 2 log|x| (power rule). The second is the square of log(x). They are completely different.
Dropping the base when applying rules. All logarithms in a calculation must have the same base before you can combine them. You cannot add log₂(x) and log₃(x) directly — convert one using the change of base formula first.
Helpful Tips
When you are stuck, convert the logarithmic equation to exponential form. This simple step clears up most confusion.
For mental math, think in powers. log₂(64): "2 to what power is 64?" Count: 2, 4, 8, 16, 32, 64 — that is 6 doublings, so the answer is 6.
If you want to verify your logarithm work or see every step of a log equation solved, scan the problem with Solver AI for an instant, detailed breakdown.
Practice Problems
Evaluate log₄(64).
Answer: 4³ = 64, so log₄(64) = 3.
3Simplify log₅(25) + log₅(5).
Answer: log₅(25 · 5) = log₅(125) = 3 (since 5³ = 125).
3Solve log₂(x − 1) = 5.
Answer: x − 1 = 2⁵ = 32, so x = 33.
Check: 33 − 1 = 32 > 0. Valid.
x = 33Solve log(x) + log(x + 3) = 1.
Answer: log(x(x + 3)) = 1, so x² + 3x = 10, giving x² + 3x − 10 = 0.
Factor: (x + 5)(x − 2) = 0. x = 2 (valid) or x = −5 (rejected, since log(−5) is undefined).
x = 2Evaluate log₇(49) − log₇(7).
Answer: log₇(49/7) = log₇(7) = 1.
1Solve 3^(2x) = 81.
Answer: 81 = 3⁴, so 3^(2x) = 3⁴, meaning 2x = 4 and x = 2.
Alternatively, take log₃ of both sides: 2x = log₃(81) = 4.
x = 2