NB: To use logarithms, all numbers must be positive, you cannot calculate the logarithm of a negative number.

In mathematics, the logarithm is the inverse operation to exponentiation e.g $2 \times 2 \times 2 = 2^3 =8$ this means that if we multiply 2 by itself 3 times the answer will be 8, this is exponentiation. The inverse process tries to figure out how many times 2 has to multiply itself to get 8, this is logarithms.

The number we are multiplying is called the “base”, so we can say:

The logarithm of 8 with base 2 is 3 or “log base 2 of 8 is 3” or “the base-2 log of 8 is 3″\\

Example 1: What is $\log_{5}{625}$ ?

We are asking “how many 5s need to be multiplied together to get 625?”

$5 \times 5 \times 5 \times 5 = 625$, so we need 4 of the 5s

Answer: $\log_{5}{625} = 4$

Example 2: What is $\log_{10}{1000}$ ?

We are asking “how many 10s need to be multiplied together to get 1 000?”

$10 \times 10 \times 10 = 1 000$, so we need 4 of the 5s

Answer: $\log_{10}{1000} = 3$

Logarithms are sometimes expressed to base $e$ and are called Natural Logarithms with the symbol $\ln$ . $e$ (Euler’s Number) is about 2.71828

It is how many times we need to use $e$ in a multiplication, to get our desired number.

Example: $\ln(7.389) = log_{e}{7.389} \approx 2$ because $2.718282 \approx 7.389$

**Exercise:**

Write the following in logarithmic form.

- $4^5 = 1024$
- $3^4 = 81$
- $ 7^2 = 49$

**Laws of Logarithms**

- $\log_b MN = \log_b M + \log_b N $
- $\log_b \frac{M}{N} = \log_b M {-} \log_b N $
- $\log_b M = \log_b N$ if and only if $M = N$
- $\log_b M^k = k \log_b M $
- $\log_b b = 1 $
- $\log_b 1 = 0 $
- $\log_b b^k = k$
- $b\log_b x = x$

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