Mathematics can sometimes be confusing, especially with terms and symbols like "log" and "ln". These terms refer to operations that calculate the logarithm of a number, but they're not identical. This article will explain the differences between log and ln with easy-to-understand examples
What is a Logarithm?
Before we talk about the differences between log and ln, let's understand what a logarithm is. A logarithm is a math operation that tells us what power we need to raise a certain number (called the base) to get another number (called the argument). This is written as log_b(x) = y, where b is the base, x is the argument, and y is the power.
For example, if we have log_10(100) = 2, it means that 10 raised to the power of 2 gives us 100 (10^2 = 100).
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What is Log?
"Log" is short for logarithm and is written as log. In math, logs can have different bases, but the most common base is 10. When the base is 10, it's called a common logarithm. The equation for a common logarithm is log_10(x) = y, where x is the argument and y is the power.
For instance, if we take log_10(100) = 2, it means that 10 raised to the power of 2 equals 100 (10^2 = 100).
What is Ln?
The term "Ln" stands for natural logarithm and is written as ln. It's a type of logarithm where the base is a special number called e, which is about 2.71828. The natural logarithm equation looks like this: ln(x) = y, where x is the number you're taking the logarithm of, and y is the result.
For example, if you have e raised to the power of 2, written as , the natural logarithm of that equals 2. In simpler terms, .
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Differences Between Log and Ln
Now, let's compare log and ln:
Base
The main difference is the base of the logarithm. Log (short for logarithm) uses base 10, while ln (natural logarithm) uses base e.
Applications
Logarithms with base 10 (log) are common in things like the Richter scale, which measures earthquake strength. Natural logarithms (ln) are more common in advanced math like calculus.
Range of Values:
Logarithms come in two types: common logarithms (log) and natural logarithms (ln). Logarithms can range from negative infinity to positive infinity. However, ln, which uses the base e, is more linked to how things grow or shrink over time, like in calculus.
Calculation:
When you calculate with log, you have to say which base you're using. But ln always uses base e. So, even if you use the same number, log and ln can give different answers. For example, log base 10 of 100 is 2, while ln of 100 is about 4.60517.
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Graphs:
Logarithms are like the opposite of exponentials. The graph of a logarithm is a mirror image of an exponential graph flipped over the line y=x.
The graph of ln is a bit like log, but it rises more slowly and gets close to a straight up-and-down line as the number it looks at gets closer to zero.
Frequently Asked Questions
A log usually means a logarithm with a base of 10. Ln refers to a logarithm with a base of e, which is also called a common logarithm or natural logarithm
To convert from log10 to ln, either multiply by M_LN10 or divide by M_LOG10E (The values are the reciprocal of the other
Logarithms are often shown in base-10, while natural logarithms (ln) use base e, where e is approximately 2.71828. When e is raised to the power of 2.303, it equals 10. Therefore, ln 10 equals 2.303. To convert ln to log, we multiply by 2.303.
The exponential function, exp : R → (0,∞), is the inverse of the natural logarithm, that is, exp(x) = y ⇔ x = ln(y). Remark: Since ln(1) = 0, then exp(0) = 1. Since ln(e) = 1, then exp(1) = e.