# 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.

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## 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 $y=\log_2(x)$