# Condense Logarithms Calculator

Welcome to Omni's **condense logarithms calculator**, where we'll see how to rewrite logarithms or rather logarithmic expressions as a single logarithm. To be precise, we'll try simplifying logs by applying **three simple formulas**. In fact, we'll use the same ones that work for **expanding logarithms**; we'll simply do it all backward. As such, we'll find out **how to subtract and how to add logs** (with the same base, mind you).

## What is a logarithm?

With **the rate at which the number of cases increases over time**. Mathematically speaking, such a thing is called exponential growth.

As you might have noticed, the name suggests that it has something to do with **exponents**. Indeed, if we assume that each infected person transmits the disease onto, say, `4`

others, then patient zero got `4`

people sick. In turn, they got `4² = 4 * 4 = 16`

people infected, who later got `4³ = 4 * 4 * 4 = 64`

people infected. In general, **we describe the number of sick** in the `n`

-th step **by the exponent** `4ⁿ`

.

**The logarithm** is the inverse function to the exponential one. To make it all precise, let's see the following log definition.

💡 `logₐ(b)` gives you the power to which you'd need to raise `a` in order to obtain `b` . Note, however, that in general, this can be [a fractional exponent](calc:1503). |

In the above epidemic example, the logarithm (with base `4`

) returns **at which step we get a fixed number of infected**. For instance:

`log₄64 = 3`

Note that taking a root is also considered an inverse operation to taking a power. However, as opposed to logarithms, **roots return the exponent base**, not the exponent itself (in the above language: they return how many people get infected by a single person). For example:

`³√64 = 4`

Before we learn how to rewrite logs, let's mention **a few critical facts** concerning them.

- There are
**two very special cases of the logarithm**which have unique notation: the natural logarithm and the logarithm with base`10`

. We denote them`ln(x)`

and`log(x)`

(the second one simply without the small`10`

), and their bases are, respectively, the Euler number`e`

and (surprise, surprise!) the number`10`

. There is also the binary logarithm, i.e., log with base`2`

, but it's not as common as the first two. **The logarithm function is defined only for positive numbers.**In other words, whenever we write`logₐ(b)`

, we require`b`

to be positive.- Whatever the base,
**the logarithm of**`1`

**is equal to**`0`

. After all, whatever we raise to power`0`

, we get`1`

. **Logarithms are extremely important.**And we mean**EXTREMELY**important. Outside of mathematics, they're used in**statistics**(e.g., the lognormal distribution),**economics**(e.g., the GDP index),**medicine**(e.g., the QUICKI index), and**chemistry**(e.g., the half-life decay). Also, quite**a few physical units**are based on logarithms, for instance, the Richter scale, the pH scale, and the dB scale.

Alright, that should be enough of a description for now. It's time to get back to **simplifying logs using concrete formulas**. We'll take on some gruesome expressions that involve logs and learn to **write the expressions as a single logarithm**.

## How to rewrite logarithms?

As we've mentioned in **the inverse operation to exponents**. Therefore, it should come as no surprise that the properties of the two are quite connected: in particular to multiplying and dividing exponents.

Let us begin by mentioning one log property that our condense logarithms calculator **doesn't** use - **the change of base formula**. It allows us to switch from one log basis to another, but at a price - we get a quotient of two expressions from a single one. If you'd like to see the specifics, make sure to check out Omni's dedicated change of base calculator.

We will, however, use here **three other formulas**. We often use them for expanding logarithms, but there's no harm in working them the other way round: for **condensing logs** instead. Still, let us see them in their original form.

- The logarithm of a product is a sum of logarithms.

- The logarithm of a quotient is a difference of logarithms.

- The logarithm of an exponent is a multiple of a logarithm.

As you can see, all of them take a single log (of a product, quotient, or exponent) and **expand it into a longer expression**. However, if we look at them backward, they write the expression as a single logarithm. That way, we obtain **formulas for adding logs, subtracting logs, and multiplying logs by a number**. The three combined are what the condense logarithms calculator is all about.

## How to condense logarithms?

We now take the theory on **work its magic to our needs**. Mind you, we'll keep to formulas for now, and once we have them in full glory, we'll move on to condensing logs examples in .

First of all, we take on the simplest of the expanding formulas: that for **a logarithm of an exponent**. Let's turn it around, fix the notation to suit the one used in the condense logarithms calculator, and have it neatly here for future use:

We'll now use it (together with the product property) to learn how to add logs. And we don't mean just any sum - we mean **adding logs with multiples**.

`x * logₙ(a) + y * logₙ(b) = ?`

We begin by applying the first condensing logs formula to both summands, i.e., we take `x`

and `y`

and drag them inside:

`x * logₙ(a) + y * logₙ(b) = logₙ(aˣ) + logₙ(bʸ)`

.

Now, with no multiples in front of the expressions, we're able to **go from adding logs to a log of a product**:

Well, we've seen how to add logs, so it shouldn't be too difficult to go from there to subtracting logs, right?

`x * logₙ(a) - y * logₙ(b) = ?`

Indeed, it isn't. We start the same: by dragging `x`

and `y`

inside,

`x * logₙ(a) - y * logₙ(b) = logₙ(aˣ) - logₙ(bʸ)`

,

and we finish by translating **subtracting logs into a log of a quotient**.

Voilà! These three are all the formulas we'll need for simplifying logs. So let's now **switch from symbols to numbers** and get an example going with the condense logarithms calculator at hand.

## Example: using the condense logarithms calculator

With all the explanations we've seen so far, there's no need for any introduction - **let's simply work out an example**. Although technically, we've just done an introduction, haven't we? Oh, bother...

We'll show how to condense the logarithms in:

`3 * log₆4 + log₆9`

Before we write the expression as a single logarithm ourselves, let's see **how to rewrite the logs with the condense logarithms calculator**.

At the top of our tool, we choose **the type of operation we're dealing with**. In our case, we have a sum, so we choose "*adding logs*" under "*Use the formula for*." That shows us a symbolic expression with the notation used underneath: `x * logₙ(a) + y * logₙ(b)`

. Comparing it to ours, we see that we need to input:

`x = 3`

, `a = 4`

, `b = 9`

, and `n = 6`

.

However, this still leaves `y`

. Recall that by convention, no number in front of a function (in this case, a logarithm, but it's the same for, say, trigonometric functions) means that the number is, in fact, equal to `1`

. Nevertheless, we don't need to input `y = 1`

since the condense logarithms calculator will **understand a blank field as** `y = 1`

.

Once we give all the necessary data, **the tool will spit out the answer** underneath together with a step-by-step application of the formulas from and a numerical approximation of the result. However, before we reveal it to the world, **let's describe how to add the logs ourselves**.

Firstly, just as we did in `3`

from in front of the log inside using the exponent property:

`3 * log₆4 + log₆9 = log₆(4³) + log₆9 = log₆64 + log₆9`

.

Next, we use the formula for how to add logs and get

`3 * log₆4 + log₆9 = log₆64 + log₆9 = log₆(64 * 9) = log₆576 ≈ 3.54741`

.

In general, **logarithms can be challenging to compute**, so we get approximations like the one above with external tools - something like the condense logarithms calculator. Or any of Omni's - **be sure to check them out**!