---
# System prepended metadata

title: 5.3 - Logarithm Functions
tags: ['5.3']

---

# 5.3 - Logarithm Functions


- The expression $$\log_a(x)=y$$ means $$a^y=x$$.



- $\log_a(x)=$'the logarithm of x to base a' or just 'the log of x to base a'.

- Typically the log notation is introduced if we want to solve the equation $a^y=x$. Solve $y=\log_a(x)$ and use a calculator to find the logarithm.

- The word logarithm comes from a combination of Greek words `logos` (ratio) and `arithmos` (number).

- Abbreviations:
$\log(x)=\log_{10}(x)$
$\ln(x)=\log_e(x)$

### Example 1. 


::: spoiler
<summary> Solution: </summary>



:::


## Examples.

Convert to a log equation:

$2^x=7$

$4^t=72$

$e^3=x$

$10^4=10000$

$7^y=x$


| #  | Exponential Equation  | Logarithmic Equation        |
|----|-----------------------|-----------------------------|
| 1  | $2^x = 7$             | $x = \log_2(7)$             |
| 2  | $4^t = 72$            | $t = \log_4(72)$            |
| 3  | $e^3 = x$             | $3 = \ln(x)$   |
| 4  | $10^4 = 10000$        | $4 = \log_{10}(10000)$      |
| 5  | $7^y = x$             | $y = \log_7(x)$             |



Convert to an exponential equation:

$\log_4(16)=2$

$\log_3(x)=3$

$\log (.00001)=-5$

$\log_2(7)=r$

$\ln(5)=-y$

| #  | Logarithmic Equation   | Exponential Equation        |
|----|------------------------|-----------------------------|
| 1  | $\log_4(16) = 2$       | $4^2 = 16$                  |
| 2  | $\log_3(x) = 3$        | $3^3 = x$                   |
| 3  | $\log (0.00001) = -5$  | $10^{-5} = 0.00001$         |
| 4  | $\log_2(7) = r$        | $2^r = 7$                   |
| 5  | $\ln(5) = -y$          | $e^{-y} = 5$                |



Evaluate.

(Let $x=$ the answer and convert to an exponential equation to find $x$.)

$\log_2(16)=$

$\log_3(27)=$

$\log_3(\frac{1}{3})=$

$\log_4(\frac{1}{16})=$

$\log_7(1)=$

$\log_{10}(.001)=$

$\log_3(\sqrt{3})=$

$\log_5(5^7)=$

$\ln(e^6)=$

$\log_4(0)=$

$\log_2(-4)=$


$\log_4(16)=$

$\log_5(\frac{1}{25})$

$\log_6(0)=$

$\log_{10}(.0000001)=$

$\log_5(-25)=$

$\log_7(7^4)=$

$\ln(1)=$

$\log_6(-1)=$

$\log_{10}(.00001)=$

$\log_6(\sqrt{6})=$


| #  | Logarithmic Equation               | Exponential Equation        | Answer ($x$) |
|----|------------------------------------|-----------------------------|--------------|
| 1  | $\log_2(16) = x$                   | $2^x = 16$                  | $x = 4$      |
| 2  | $\log_3(27) = x$                   | $3^x = 27$                  | $x = 3$      |
| 3  | $\log_3\left(\frac{1}{3}\right) = x$| $3^x = \frac{1}{3}$         | $x = -1$     |
| 4  | $\log_4\left(\frac{1}{16}\right) = x$| $4^x = \frac{1}{16}$        | $x = -2$     |
| 5  | $\log_7(1) = x$                    | $7^x = 1$                   | $x = 0$      |
| 6  | $\log_{10}(0.001) = x$             | $10^x = 0.001$              | $x = -3$     |
| 7  | $\log_3(\sqrt{3}) = x$             | $3^x = \sqrt{3}$            | $x = \frac{1}{2}$ |
| 8  | $\log_5(5^7) = x$                  | $5^x = 5^7$                 | $x = 7$      |
| 9  | $\ln(e^6) = x$                     | $e^x = e^6$                 | $x = 6$      |
| 10 | $\log_4(0) = x$                    | N/A (undefined)             | undefined    |
| 11 | $\log_2(-4) = x$                   | N/A (undefined for negative)| undefined    |
| 12 | $\log_4(16) = x$                   | $4^x = 16$                  | $x = 2$      |
| 13 | $\log_5\left(\frac{1}{25}\right) = x$| $5^x = \frac{1}{25}$        | $x = -2$     |
| 14 | $\log_6(0) = x$                    | N/A (undefined)             | undefined    |
| 15 | $\log_{10}(0.0000001) = x$         | $10^x = 0.0000001$          | $x = -7$     |
| 16 | $\log_5(-25) = x$                  | N/A (undefined for negative)| undefined    |
| 17 | $\log_7(7^4) = x$                  | $7^x = 7^4$                 | $x = 4$      |
| 18 | $\ln(1) = x$                       | $e^x = 1$                   | $x = 0$      |
| 19 | $\log_6(-1) = x$                   | N/A (undefined for negative)| undefined    |
| 20 | $\log_{10}(0.00001) = x$           | $10^x = 0.00001$            | $x = -5$     |
| 21 | $\log_6(\sqrt{6}) = x$             | $6^x = \sqrt{6}$            | $x = \frac{1}{2}$ |


###  Graph of $y=\log_2(x)$

| $x$              | Work                                         | $y = f(x) = \log_2(x)$ |
|------------------|----------------------------------------------|------------------------|
| $2$              | $2^y = 2 \implies y = 1$                     | $1$                    |
| $4$              | $2^y = 4 \implies 2^2 = 4 \implies y = 2$    | $2$                    |
| $8$              | $2^y = 8 \implies 2^3 = 8 \implies y = 3$    | $3$                    |
| $1$              | $2^y = 1 \implies 2^0 = 1 \implies y = 0$    | $0$                    |
| $\frac{1}{2}$    | $2^y = \frac{1}{2} \implies 2^{-1} = \frac{1}{2} \implies y = -1$ | $-1$                   |
| $\frac{1}{4}$    | $2^y = \frac{1}{4} \implies 2^{-2} = \frac{1}{4} \implies y = -2$ | $-2$                   |
| $\frac{1}{8}$    | $2^y = \frac{1}{8} \implies 2^{-3} = \frac{1}{8} \implies y = -3$ | $-3$                   |
| $\frac{1}{16}$   | $2^y = \frac{1}{16} \implies 2^{-4} = \frac{1}{16} \implies y = -4$ | $-4$                   |
| $0$              | $\log_2(0)$ is undefined                      | Undefined              |
| $-2$             | $\log_2(-2)$ is undefined                     | Undefined              |
| $-4$             | $\log_2(-4)$ is undefined                     | Undefined              |


Graph of $y=\log_2(x)$

![{F6989490-2527-4593-824F-AD91DEDC0844}](https://hackmd.io/_uploads/H1VDPSNAR.png)

