Contact The Learning Centre

# Introduction to Logarithms

### Introduction to Logarithms

• The logarithm of a number is the exponent to which the base must be raised to produce that number.
• Example: How many $$3$$’s do we multiply to get $$81$$? Answer:$$3 \times 3 \times 3 \times 3 = 3^4 = 81$$. So the logarithm to the base $$3$$ of $$81$$ is $$4$$.
• We write "the number of $$3$$’s you need to multiply to get $$81$$ is $$4$$" as $$\log_{3}(81)=4$$. So these two things are equivalent:

$3 \times 3 \times 3\times 3= 3^4= 81 \quad\Longleftrightarrow\quad \log_{3} (81) = 4$

• The number we are multiplying is called the "base", so we would say: "the logarithm of 81 with base 3 is 4" or "logarithm of 81, to the base 3, is 4".
• For example: what is $$\log_5(625)$$? In this case we are asking, "how many $$5$$’s need to be multiplied together to get $$625$$?"

$5 \times 5 \times 5 \times 5 = 5^{4} = 625$

so we need $$4$$ lots of $$5$$’s. Therefore, the answer is:

$\log_{5}(625) = 4$

• We write the logarithm to the base $$10$$ as $$\log_{10} = \log$$.
• There is a special case for logarithms, when we use the irrational number, $$e\approx2.718281\ldots$$ as the base. When using this base, $$\log_e$$ we say the natural logarithm and can use the shorter notation of $$\log_e=\ln$$.