A common logarithm is a logarithm with base 10, written as log(x) or log10(x). In Intermediate Algebra, it tells you what power of 10 gives a number.
A common logarithm in Intermediate Algebra is a logarithm with base 10. When you see log(x), it usually means “what power of 10 gives x?” So log(1000) = 3 because 10^3 = 1000.
That idea is the whole point of the notation. Exponent form and logarithmic form say the same thing, just from different angles. If 10^y = x, then log(x) = y. This is why logarithms show up right after exponent rules, because they are built from the same relationship between a base, an exponent, and a result.
The common logarithm has a few anchor values you should know cold. log(1) = 0 because 10^0 = 1, and log(10) = 1 because 10^1 = 10. These are easy checkpoints when you are evaluating expressions or checking whether your answer makes sense.
In Intermediate Algebra, the common log is usually written without the base, but the base is still understood to be 10. That shorthand saves space, yet it can confuse people who are expecting every logarithm to look explicit. If the base is not written, this course usually assumes base 10 unless the problem says otherwise.
The graph of the common logarithm is the graph of a logarithmic function with base 10. It only exists for positive x-values, and it passes through (1, 0). As x gets closer to 0 from the right, the graph drops without bound, which matches the rule that you cannot take the log of 0 or a negative number.
You also meet common logarithms when you simplify expressions using logarithm properties. Since logs turn multiplication into addition and powers into multiplication, they are a cleaner tool for some algebra problems than raw exponent rules alone.
Common logarithm shows up right where Intermediate Algebra shifts from ordinary equations to exponential and logarithmic functions. Once you know that log means “what exponent goes with 10,” you can move between exponential form and logarithmic form instead of treating them as two separate topics.
That matters for solving equations. A problem like 10^x = 2500 becomes manageable when you rewrite it as x = log(2500). Even if your class uses a calculator for the decimal value, you still need the setup to be correct before you press anything.
It also matters for graphing. The parent behavior of y = log(x) gives you the basic shape, the vertical asymptote at x = 0, and the point (1, 0). Once you know that, you can compare it to other functions that are stretched, compressed, shifted, or reflected.
Common logarithms also connect directly to the properties of logarithms in the next section of the unit. When you rewrite log of a product, quotient, or power, you are using the same base-10 structure. So this term is not just a label, it is the entry point for solving, simplifying, and interpreting log expressions throughout the chapter.
Keep studying Intermediate Algebra Unit 10
Visual cheatsheet
view galleryLogarithmic Function
A common logarithm is one specific logarithmic function with base 10. If you understand the general form of a logarithmic function, then common logarithm is the version your class uses most often when the base is not written. It is the same family of function, just with a fixed base.
Base-10 Logarithm
Base-10 logarithm is another name for common logarithm, so these terms usually mean the same thing. The phrase makes the base explicit, while log(x) is the shortcut notation you will see most often in algebra. If a problem says “base-10 log,” you should think log10(x) or just log(x).
Exponential Function
Common logarithms and exponential functions are inverse operations. That means one undoes the other, so an exponential equation can often be rewritten as a logarithmic equation. In Intermediate Algebra, this connection is what lets you solve for an exponent when the base is 10.
Parent Function
The graph of y = log(x) is the parent function for common logarithmic graphs. It gives you the standard shape, intercept pattern, and vertical asymptote before any shifts or stretches are added. When a graph changes, you compare it back to this parent form.
A quiz or problem set will usually ask you to evaluate a common logarithm, rewrite an exponential equation in log form, or graph y = log(x) using key points and the asymptote. You may also be asked to decide whether an expression is defined, which means checking that the input is positive. For example, log(100) is valid, but log(0) and log(-3) are not.
When the question is computational, the setup matters more than the decimal answer. If you can see that 10^2 = 100, then you know log(100) = 2 right away. For graphing, look for the point (1, 0), the vertical asymptote at x = 0, and the increasing shape of the parent graph. Those are the features teachers usually expect you to identify quickly.
Common logarithm uses base 10, while natural logarithm uses base e. They are both logarithms and both answer the same kind of question about exponents, but the base changes the value and the notation. In Intermediate Algebra, log usually means base 10 unless the problem specifically says ln.
Common logarithm means base 10, so log(x) asks what power of 10 gives x.
log(1) = 0 and log(10) = 1 are the two anchor values worth memorizing first.
A common log is only defined for positive inputs, so log(0) and log of a negative number are not real-valued in this course.
Common logarithms are the inverse of exponential functions, which makes them useful for rewriting and solving equations.
The graph of y = log(x) has a vertical asymptote at x = 0 and passes through (1, 0).
A common logarithm is a logarithm with base 10. It tells you the exponent you need on 10 to get a given number, so log(1000) = 3 because 10^3 = 1000. In algebra, this is the default meaning of log(x) unless another base is written.
log usually means base 10, while ln means base e. They are both logarithms, but they use different bases, so they do not give the same values. In Intermediate Algebra, if the base is not shown and the problem is about common logarithms, log means base 10.
Look for the exponent that makes 10 turn into the number inside the log. For example, log(100) = 2 because 10^2 = 100. If the number is not a perfect power of 10, you may use a calculator or rewrite it using log properties.
Because no real power of 10 gives 0 or a negative result. Exponential expressions with base 10 are always positive, so the inverse log function only accepts positive inputs. That is why the domain of y = log(x) is x > 0.