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Logarithms reverse exponentiation, connecting formal definitions, exponential forms, and growth problems that ask for an unknown power.

---

## Understanding Logarithms

Logarithm is a mathematical operation that is the inverse of exponentiation. If we have an exponential equation $$b = a^c$$, then its logarithmic form is $$^a\log b = c$$.

Visible text: Logarithm is a mathematical operation that is the inverse of exponentiation. If we have an exponential equation , then its logarithmic form is .

### Formal Definition of Logarithms

Let $$a$$ be a positive number where $$0 < a < 1$$ or $$a > 1$$, and $$b > 0$$, then:

Visible text: Let be a positive number where or , and , then:

```math
^a\log b = c \text{ if and only if} b = a^c
```

Where:

- $$a$$ is the base of the logarithm
- $$b$$ is the number whose logarithm we are finding (numerus)
- $$c$$ is the result of the logarithm

Visible text: - is the base of the logarithm
- is the number whose logarithm we are finding (numerus)
- is the result of the logarithm

We can read $$^a\log b = c$$ as: $$a$$ raised to what power equals $$b$$? The answer is $$c$$. Because $$a^c = b$$.

Visible text: We can read as: raised to what power equals ? The answer is . Because .

## Relationship Between Exponents and Logarithms

Logarithms and exponents are related as operations that are inverses of each other. Consider the following examples:

| Exponential Form                            | Logarithmic Form                               |
| ------------------------------------------- | ---------------------------------------------- |
| $$2^5 = 32$$              | $$^2\log 32 = 5$$            |
| $$3^2 = 9$$               | $$^3\log 9 = 2$$             |
| $$5^{-2} = \frac{1}{25}$$ | $$^5\log \frac{1}{25} = -2$$ |
| $$7^0 = 1$$               | $$^7\log 1 = 0$$             |

Visible text: | Exponential Form | Logarithmic Form |
| ------------------------------------------- | ---------------------------------------------- |
| | |
| | |
| | |
| | |

## Common Logarithm (Base Ten)

Logarithm with base $$10$$ is called the common logarithm. It is often simplified by omitting the base $$10$$:

Visible text: Logarithm with base is called the common logarithm. It is often simplified by omitting the base :

```math
^{10}\log a = \log a
```

## Applications of Logarithms in Exponential Growth

### Determining Time to Reach a Specific Quantity

A bacterial colony initially consists of $$2{,}000 \text{ bacteria}$$ that divide every $$1 \text{ hour}$$. The growth of these bacteria follows an exponential function:

Visible text: A bacterial colony initially consists of that divide every . The growth of these bacteria follows an exponential function:

```math
f(x) = 2{,}000(2^x)
```

where $$x$$ is time in hours.

Visible text: where is time in hours.

Then, to determine the time needed for bacteria to reach a specific number, for example $$100{,}000 \text{ bacteria}$$, we need to find the value of $$x$$ that satisfies:

Visible text: Then, to determine the time needed for bacteria to reach a specific number, for example , we need to find the value of that satisfies:

```math
100{,}000 = 2{,}000(2^x)
```

By dividing both sides by $$2{,}000$$:

Visible text: By dividing both sides by :

```math
50 = 2^x
```

To find the value of $$x$$, we use the concept of logarithms:

Visible text: To find the value of , we use the concept of logarithms:

```math
x = \log_2 50
```

This demonstrates that logarithms are very useful tools for solving exponential equations, especially when finding the exponent value that yields a specific result.